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Edward Bright Mathematics Dept THE APPLICATIONS OF ELLIPTIC FUNCTIONS. THE APPLICATIONS OF ELLIPTIC FUNCTIONS BY ALFRED GEORGE GREENHILL M.A., F.R.S., PROFESSOR OK MATHEMATICS IN" THE ARTILLERY COLLEGE, WOOLWICH MACMILLAN AND CO. AND XETT YORK 1892 [All riyhlft reserved] AY* CONTENTS. PAGE INTRODUCTION, vii CHAPTER I. THE ELLIPTIC FUNCTIONS, - - - 1 CHAPTER II. THE ELLIPTIC INTEGRALS, - - 30 CHAPTER III. GEOMETRICAL AND MECHANICAL ILLUSTRATIONS OF THE ELLIPTIC FUNCTIONS, - 66 CHAPTER IV. THE ADDITION THEOREM FOR ELLIPTIC FUNCTIONS, - - - - 112 CHAPTER V. THE ALGEBRAICAL FORM OF THE ADDITION THEOREM, 142 CHAPTER VI. THE ELLIPTIC INTEGRALS OF THE SECOND AND THIRD KIND, - - 175 CHAPTER VII. THE ELLIPTIC INTEGRALS IN GENERAL AND THEIR APPLICATIONS, - 200 CHAPTER VIII. THE DOUBLE PERIODICITY OF THE ELLIPTIC FUNCTIONS, - - - 254 781468 v i CONTENTS. CHAPTER IX. PAGE THE RESOLUTION OF THE ELLIPTIC FUNCTIONS INTO FACTORS AND SERIES, ... 277 CHAPTER X. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS, - - - 305 APPENDIX, - - 340 INDEX, 353 INTRODUCTION. "L ETUDE approfondie de la nature est la source la plus feconde des decouvertes mathematiques. Non seulement cette etude, en offrant aux recherches un but determine , a Favantage d exclure les questions vagues et les calculs sans issue ; elle est encore un moyen assure de former 1 Analyse elle-meme, et d en decouvrir les elements qu il nous importe le plus de connaitre et que cette science doit toujours conserves Ces ele ments fondamentaux sont ceux qui se reproduisent dans tous les effets naturels." (Fourier.) These words of Fourier are taken as the text of the present treatise, which is addressed principally to the student of ApjDlied Mathematics, who will in general acquire his mathe matical equipment as he wants it for the solution of some definite actual problem ; and it is in the interest of such students that the following Applications of Elliptic Functions have been brought together, to enable them to see how the purely analytical formulas may be considered to arise in the discussion of definite physical questions. The Theory of Elliptic Functions, as developed by Abel and Jacobi, beginning about 1826, although now nearly " *-~~^^*~ seventy years old, has scarcely yet made its way into the viii THE APPLICATIONS OF ELLIPTIC FUNCTIONS. ordinary curriculum of mathematical study in this country ; and is still considered too advanced to be introduced to the student in elementary text-books. In consequence of this omission, many of the most interest ing problems in Dynamics are left unfinished, because the complete solution requires the use of the Elliptic Functions ; these could not be introduced without a long digression, unless a considerable knowledge is presupposed of a course of Pure Mathematics in this subject. But by developing the Analysis as it is required for some particular problem in hand, the student of Applied Mathe matics will obtain a working knowledge of the subject of Elliptic Functions, such as he would probably never acquire from a study of a treatise like Jacobi s Fundamenta Nova, where the formulas are established and the subject is developed in strictly logical order as a branch of Pure Mathematical Analysis, without any digression on the application of the formulas, or on the manner in which they originate independently, as the expression of some physical law. In introducing these applications we are following, to some extent, the plan of Durege s excellent treatise on Elliptic Functions (Leipsic, Teubner); and also of Halphen s Traite des fonctions elliptiqucs et de leurs applications (Paris, 1886-1891). But while volume I. of Halphen s treatise is devoted entirely to the establishment of the formulas and analytical properties of the functions, and the applications are not discussed till volume II. ; in the following pages it is proposed to develop the formulas immediately from some definite physical or geometrical problem ; and the reader who wishes to follow up the purely analytical development of the subject is referred to such treatises as Abel s (Euvres, Jacobi s Fundamenta Nova, INTRODUCTION. i x already mentioned, or the Treatises on Elliptic Functions of Cayley, Enneper, Kb nigsberger, H. Weber, etc. The following works also may be mentioned as having been consulted in the preparation of this work : Legendre: Theorie des f auctions elliptiques ; 1825. Thomas : Abriss einer Theorie der complexen Functionen und der Tketafunctionen einer Verdnderlichen ; 1873. Schwarz: Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen. Klein (Morrice) : Lectures on the Icosahedron ; 1888. Klein und Fricke; Vorlesungen uber die Theorie der ellip tischen Modalfunctionen ; 1890. Despeyrous et Darboux: Cours de niecanique ; 1886. R A. Roberts : Integral Calculus ; 1887. Bjerknes: Niels Hendrik Abel; tableau de sa vie et de son action scientifique ; 1885. We shall begin by the discussion of the Problem of the , as the problem best calculated to define the Elliptic Functions, and to give the student an idea of their nature and importance. Previously to the introduction of the Elliptic Functions, the Circular Pendulum could only be treated by means of the circular functions, by considering the oscillations as indefinitely small, and by assimilating its motion to that of Huygens Cycloidal Pendulum, of 1673. But now the employment of the Elliptic Functions renders the ordinary discussion of the Cycloidal Pendulum antiquated and of mere historical interest, and banishes from our treatises such expressions as " an integral which cannot be found," or "reducible to a matter of quadrature" in describing an elliptic integral, expressions which aroused the indignation of Clifford Mathematical Papers, p. 562). x THE APPLICATIONS OF ELLIPTIC FUNCTIONS. According to the new regulations for the Mathematical Tripos at Cambridge, to come into force in the examination in May 1893, the schedule II. of Part I. includes " Elementary Elliptic Functions, excluding the Theta Functions and the theory of Transformation " ; so it is to be hoped that this reintroduction of Elliptic Functions into the ordinary mathe matical curriculum will cause the subject to receive more general attention and study. These Applications have been put together with the idea of covering this ground by exhibiting their practical importance in Applied Mathematics, and of securing the interest of the student, so that he may if he wishes follow with interest the analytical treatises already mentioned. We begin with Abel s idea of the inversion of Legendre s elliptic integral of the first kind, and employ Jacobi s notation, with Gudermann s abbreviation, for a considerable extent at the outset. The more modern notation of Weierstrass is introduced subsequently, and used in conjunction with the preceding notation, and not to its exclusion ; as it will be found that sometimes one notation and sometimes the other is the more suitable for the problem in hand. At the same time explanation is given of the methods by which a change from the one to the other notation can be speedily carried out. It has been considered sufficient in many places, for instance in the reduction of the Integrals in Chapter II., to write down the results without introducing the intermediate analysis ; as the trained mathematical student to whom this book is addressed will have no difficulty in supplying the connecting steps, and this work will at the same time provide instructive exercises in the subject ; and further, in the interest of such students, many important problems have been introduced in INTRODUCTION. XI the text, forming immediate applications of theorems already developed previously. I have to thank Mr. A. G. Hadcock for his assistance in preparing the diagrams, and in drawing them carefully to scale. ERRATA. Page 6. Line 9 from bottom, read Huygeiis. 42. Line 6, read sin" 1 */ -. V x-y 48. Line 5 from bottom, read - 4tt-(9e- -r 4/r) J . 64. Line 19, read Fonctions elliptiques. 99. The diagram must be replaced by the one given below. The Xodoid in fig. 12, p. 99, was described by a point which was not a focus of the rolling hyperbola. 107. Line 2 from bottom, delete minus sign before radical. 138. Equation (7), read (r., 2 - ctf/D. 158. Line 12, read 3QX(x, y). 205. Line 6 from bottom, read $(u - v) - $>(u + v). 213. Line 7 from bottom, read G + Lx - X(yz - y ~] - with the corresponding subsequent corrections. 227. Line 7, read P s /- Y i + Q\ ; - Y 2 = - 282. Line 5 from top, for rectangle read ribbon. 328. Line 12 from bottom, read Pw. L. M. .?., IX. ABBREVIATIONS. Q. J. M., Quarterly Journal of Mathematics. Proc. L. M. S., Proceedings of the London Mathematical Society. Proc. G. P. $., Proceedings of the Cambridge Philosophical Society. Am. J. M., American Journal of Mathematics. F. E., Fonctions elliptiques (Legendre and Halphen). Math. Ann., Mathematische Annalen. Phil. Mag., Philosophical Magazine. Phil. Trail*. Philosophical Transactions of the Royal Society of London. Berlin Sitz., Sitzungsberichte der Berliner Akademie. CHAPTEK I. THE ELLIPTIC FUNCTIONS. 1. The Pendulum; introducing Elliptic Functions into Dynamics. When a pendulum OP swings through a finite angle about a horizontal axis 0, the determination of the motion introduces the Elliptic Functions in such an elementary and straight forward manner, that we may take the elliptic functions as defined by pendulum motion, and begin the investigation of their use and theory by their application to this problem. Denote by W the weight in Ib. of the pendulum, and let OG = h (feet), where G is the centre of gravity ; let Wk 2 denote the moment of inertia of the pendulum about the horizontal axis through G, so that W(h* + k 2 ) is the moment of inertia about the parallel axis through (fig. 1). Then if OG makes with the vertical OA an angle 6 radians at the time t seconds, reckoned from an instant at which the pendulum was vertical ; and if we employ the absolute unit of force, the poundal, and denote by g (32 celoes, roughly) the acceleration of gravity, the equation of motion obtained by taking moments about is since the impressed force of gravity is Wg poundals, acting vertically through G ; so that or, on putting h + k*/h = I, G.E.F. THE ELLIPTIC FUNCTIONS. to THE ELLIPTIC FUNCTIONS. 3 If the gravitation unit of force, the force of a pound, is employed, then the equation of motion is written w J lft fr/TO . 7 O\*-^ v/ TTTf " /\ (A. 2 +& 2 )-p = Wh sm 0, reducing to (1) as before. 2. Producing OG to P, so that OP = l, GP = k 2 /h, the point P is called the centre of oscillation (or of percussion) ; and is called the length of the simple equivalent pendulum, because the point P oscillates on the circle AP in exactly the same manner as a small plummet suspended by a fine thread from (fig. 2); as is seen immediately by resolving tangentially along the arc AP = s = l9 ) when the equation of motion of the plummet is = g sin#= or I(d 2 0/dt 2 )= -g sin#; ........................ (1) and integrating, U(dO/dt) 2 = C-gversO ......................... (2) These theorems are explained in treatises on Analytical Mechanics, such as Kouth s Rigid Dynamics, or Bartholomew Price s Infinitesimal Calculus, vol. IV., and might have been assumed here ; but now we proceed further, to the complete integration of equation (2). 3. First suppose the pendulum to oscillate, the angle of oscillation BO A +AOB being denoted by 2a (fig. 2); the angle of oscillation is purposely made large, as in early clocks, in the Navez Ballistic Pendulum, in a swing, or as in ringing a church bell, so as to emphasize the difference from small oscillations, the only case usually considered in the text books ; in fig. 2 the angle of oscillation is made 300. Then dO/dt = when 6 = a, so that in equation (2) (7=# versa ; and now denoting g/l by n 2 , so that n is what Sir W. Thomson calls the speed (angular) of the pendulum, ^(dO/dt) 2 = n 2 ( vers a - vers 0) = 2?i 2 (sin 2 Ja-sin 2 ^), .............................. (3) since vers = 2 sin 2 J0 ; dO/dt = 27i x /(sin 2 ia -sin 2 i0), and nt-f ^ _ -.. . ..(4) 4, THE ELLIPTIC FUNCTIONS. and (4) is called by Legendre an elliptic integral of the first kind ; it is not expressible by any of the algebraical, circular, or hyperbolic functions of elementary mathematics. 4. To reduce this elliptic integral to the standard form con sidered by Legendre, we put sinA0 = sinJa sin 0, equivalent geometrically to denoting the angle ADQ by (fig. 2), where AQD is the circle on AD as diameter, touching BB f in D, and cutting the horizontal line PN in Q. For, in the circle AP, and, in the circle AQ, AN= \AD vers 2$ = AD sin 2 = I vers a sin 2 = 21 sin 2 |a sin 2 0. Now sin 2 \ a sin 2 J$ = sin 2 |a cos 2 0, and J$ = sin ~ 1 (sin Ja sin 0), so that dle= .^i*.* *ft, v ^/(l snrja sm 2 0) and therefore nt = fr- - r~r ^r-, > J ^/(1-siu 2 iasm 2 ^,) which is now an elliptic integral of the first kind, in the standard form employed by Legendre. (Fonctions Elli ptiques, t. I., chap VI.) 5. In Legendre s notation, sin-Jet is replaced by ic; the quantity ic sin 2 <) is. denoted by A</> or A(0, K) ; and the integral ry(l-/c 2 sin 2 ^)-^ is denoted by F<f> or I\<J>,K), and called the elliptic integral of the first kind, being called the amplitude and K the modulus. Thus, in the pendulum motion, nt^Ffj), or F(</>, sinja). Legendre employs c instead of K, and puts K = sin 6 (a different ^ to what we have just employed) and calls the modular angle ; and he has tabulated the numerical values of F((/>, K) for every degree of and 0. (Fonctions Elliptiques, t. II. Table IX.) Legendre spent a long life in investigating the properties of the function Fc/>, the elliptic integral of the first kind ; but the subject was revolutionised by the single remark of Abel (in THE ELLIPTIC FUNCTIONS. .5 1823), that F(j> is of the nature of an inverse function ; and that if we put u F(f>, then we should study the properties of 0, the amplitude, as a function of u, and not of u as a function of <j>, as carried out by Legendre in his Fonctions Elliptiques. 6. Jacobi proposed the notation = am u, or am(it, /c) when the modulus K is required to be put in evidence ; and now, considered as functions of u, we have Jacobi s notation cos < = cos am u, sin < = sin am u, A</> = A am u, the three elliptic functions of u\ and in Jacobi s Fundamenta Nova (1829) the properties of these functions, cos am u, sin am u, A am u f are developed, the elegance of Jacobi s notation tending greatly to the popularity of this treatise. 7. Definition of the Elliptic Functions. Jacobi s notation is rather lengthy, so that nowadays, in accordance with Gudermann s suggestion (Tkeorie der Modular Functionen, Crelle, t. 18), cos am u is abbreviated to en it, sin am u to sn u, and A am u to dn u ; and en u, sn u, dn u are the three elliptic functions (pronounced, according to Hal- phen, with separate letters, as c, n, u ; s, n, u ; d, n, u) ; and they are defined by en u = cos 0, sn u = sin 0, dn u = A< = ^/(l /c 2 sin 2 <) ; where ^ is a function of u, denoted by am u, and defined by the relation so that /~amtt u = A/( 1 /c 2 sin 2 0) - $d<j> ; cZam u d(j> . -> and Y - = j r ^/ (I K" sm z (p) = dnit. rp , dcnu dcosd . dd> Thence , = , = sm0-^-= snudnu: du du * du and similarly d sn u d sin c?0 7 - = 7 - = cos -f- = en u dn u ; du du ^ du n u <j> /rsn d> cos c< and , = ~r t -= ---- - ^ = /c 2 sn i^ en u du du A0 cZt6 THE ELLIPTIC FUNCTIONS. 8. Returning now with these definitions and this notation to the motion--o-~ilie^ pendulum, we have, on comparison, u = nt, while K = sin |<v\so that the modular angle is \a ; and K = AD/AB = AB/AE, K * = AD/AE (fig. 2) ; also ^ = am u, cos == en u, sin </> = sn u, d<f>/dt = n dn u ; d6/dt = 2nic cnu = 2 /i/c en nt, sin|$ = K sn u = K sn nt, cos J(9 = dn it = dn ?i ; AN = AD srfnt, ND = AD en 2 *, NE=AE dn*nt ; giving these quantities as elliptic functions of u or nt. 9. We notice that ic = for infinitely small oscillations of the pendulum, the only case usually treated in the text- books ; and now <j> = u = nt, so that en u = cos u, sn u = sin u, while dn u = 1 ; and the elliptic functions have degenerated into the ordinary circular functions of Trigonometry. But in finite oscillations of the pendulum, where K is not zero, these new functions are required, which are called the elliptic functions; and their geometrical definition is exhibited in fig. 2, in a manner similar to that employed in Trigonometry for the circular functions. The name elliptic function is somewhat of a misnomer ; but arose from the functions having been first approached by mathematicians in their attempt at the rectification of the ellipse ( 77). For finite oscillations the circular functions are applicable only to cycloidal oscillations, as discovered by HuygKens, 1673, whence the motion on the arc of a cycloid is generally investi gated at length in elementary treatises ; but this discussion may be considered as of mere antiquarian interest, now that we are proceeding to discuss the finite oscillations of the pendulum by the aid of the elliptic functions. We may however make here a slight digression on cycloidal oscillations, treated in the manner we have employed for circular oscillations. THE ELLIPTIC FUNCTIONS. 10. Gydoidal Oscillations. In the cycloid, fig. 4, the angle ADQ or <j> = nt (not &mnt, as in the circular pendulum) for all finite oscillations ; for as P oscillates on the arc BAB of the inverted cycloid described by the rolling of the circle AE, Q follows P at the same level on the circle AD with constant velocity. For if PQF meets the circle on AE as diameter in R, then, from a well-known property of the cycloid, the tangent TP is equal and parallel to A R, and half the arc AP ; and if n, p, q, r denote simultaneous consecutive positions of N, P, Q, R, the velocity of Q _ ^ Qq _ , Qq, Nn the velocity of P Pp ~~ Nn Pp = cosec qQP sin pPQ = cosec AFQ sin AER = _ IAN.AE N Now the velocity of P = ^(jlg . ND) and therefore the velocity of Q = ^AD Mti /(2g/AE) = AD^(g/l) = n . AD, a constant, if AE=\l; and therefore the angular velocity of Q about D is n, and the angle ADQ = <p = nt. Therefore the oscillations are isochronous, since the period is independent of the amplitude of oscillation. 8 THE ELLIPTIC FUNCTIONS. But in the circular pendulum the period increases with the amplitude or angle of oscillation; because in the circle AP (fig. 2) the versed "sine AN varies as the square of the chord AP, while in the cycloid AP (fig. 4) the versed sine AN varies as the square of the arc AP. The time from P to A on the cycloid is equal to the c.m. (circular measure) of the angle ADQ divided by n or +J(gjl) ; and generally the time over any finite arc Pp of the cycloid will be equal to the c.m. of the corresponding angle QDq divided by n, supposing the body to start from the level of D. This will be true even when the point D is above E, as at D f , so that the body enters the cycloid with given velocity ; as for instance in the case of a railway train entering with given velocity V a cycloidal tunnel BAB under a river. Making DD = ^V 2 /g, the impetus of the velocity V, then the time occupied by the train in the tunnel from B to B is twice the c.m. of AD C divided by n. Also if the length of the tunnel is 2s, then s = ^/(2lh), if AD, the depth or versed sine of the tunnel, is h ; so that the time occupied is 2, .DC II. , IAD 2s If h 11. The Period of the Pendulum, and of the Elliptic Functions. The ^period of the pendulum is the name now given to the time of a double swing, according to the report of a Com mittee at the Conference of Electricians in Paris, 1889 : thus, if the swing is small, the period is 27r x /(/</) seconds. But if the angle, of vibration 2a is finite, the period is in creased ; denoting the period by T, and therefore the quarter- period, or time of motion of P from A to B (fig. 2) by %T, then as t increases from to \T, increases from to a, and from to J?r, so that nt or u increases from to K, where ( 4) and K (or F I K in Legendre s notation, and called by him the complete elliptic integral of the first kind) is now called the real quarter period of the elliptic functions, to the modulus K. THE ELLIPTIC FUNCTIONS. Now, expanding by the Binomial Theorem, z =1 and, by Wallis s Theorem, /IT (sin Thus the period of a pendulum of length I, oscillating through . an angle 2a, is As a first approximation therefore in the correction for am plitude of swing, the period must be increased by the fraction J(sin |-) 2 of itself, or by 100(^ chord of a) 2 per cent. Thus a pendulum, which beats seconds when swinging through an angle of 6, will lose 11 to 12 seconds a day if made to swing through 8, and 26 seconds a day if made to swing through 10. (Simpson s Fluxions, 464.) The value of K or / V has been tabulated by Legendre for every degree and tenth of a degree in the modular angle (Fonctions Mliptiques, t. II., Table I.). We denote the modular angle by Ja, and put /c = sinja; while cosja is denoted by K arid called the complementary modulus, so that i 2 1 K- + K ~ = 1 ; and then F I K is denoted by K , and called the complementary quarter period. The following table (from Bertrand s Calcul Integral, p. 714), gives the logarithms of the quarter periods /i" and -ZiTjCorrespond- ingtoeveryhalf degree in Ja, the quarter angle of swing; and then 2/oc = sin a, /c = sinja, //^cosja, and ia is the modular angle. The modular angle in the Table is given from to 45 ; to determine K for a modular angle greater than 45, we look out the value of K corresponding to the complementary modu lar angle. 10 THE ELLIPTIC FUNCTIONS. >O O O O ip O O O >O p O p O O \O O ip O O O p O >O O (N <N (M ^ ^H eOOGOtOCC G51.-^iOO lOGOCO *C^ > lOOOOCiCiOCiOOOCOGOGOCOl^l^ rerccCC^<^<N(>lC^C^C<lC^Cv|<^C<laN 8^ 01 co cc cc ig i i CC -^ CO GO Oi --I coi o p p p >o p ip p o p o p O p p p >p O US O iO O O O O O 5 OOGQl-tODOiO-*iTt<CCCC(M<Nr-i,-<OOa5 JCOCOCDCOCOCOCOCOCOCOCQCDCOCOCDCOCOCOlO GSCOfNCOcoco-HCOOOcOCOi i c^ccr-^ccaoioccciTfiiOi iC^r 2CCO - CCO5OlO^ ^ wl O Oi O i i C<1 iO i"- O - CC O L- IO (M O CO CC t I t>. CO CC 1^ CiOOiOOirHC <MOOCDrt<^coOilOC<lO CCQO^fC^rHCCCi)^H i CO <N t^ CO O5 O i-H OO -O <M C5 CO CO i i O5 r IO CC (N -^ IO O I GO Ci CO CO i i O5 r IO CC (N O ^H(NCO ^lOCCl GO p o p ip p ip p ip p p p ip p >p p o p ip p p p o p ip p ip p ip pop ^CCi ilOOi iCCL t>-OCClO OOr- iCOCOOSiOlOL^-GMCiGC^iCCr-^i i^^i I 3 h- IQ i I CO O I>- l>- IO CC (N C>J CO Tfl IO !- Gl (>4 IO GO C-4 IO Ci ^ GO (M I C^l 1- CM CC GO CC O I IO CC (M O O5 CO I O IO * CC (M i i i O C5 O GC l^ t- CO CO UO iO Tf ^ Sl^-l>.l^.COCOCOCOCOO OiOiOiO 1 OiOOOiOiO ^ ^^ - rf-ti^-Tt<Tti rti^t(Tti CJGOCC fCNCOl^-lOClOl i I 40 ^ p p p p p p >p p ip p p p THE ELLIPTIC FUNCTIONS. 11 p P p p O p Lt O ut p ip p p O ip O O p ip p O p O O O p \~ O p O -e- cccocot^ oc:c<>cic:o-icot-ou :r:i^!M-^ccc5XOiCM < ;c M -cc r-H C4 CO IO t>> O C4 IO O) CN C 1-1 IO O CD < t> CO O t ^ ^H O> <X> CO O ^ ^1 -^ -V ^ JSOvOjCOt^t^L^I>.CO^c:C; OO^i Mrtrj-futut^t^OCCip^fN o ^ ippippippippi^pctpippiopL^oippippippopippop uioot^i -^ooocc-. OOO^H- ^I Mccrc-^^ooocbr^i^ocaociOiO rj< * -^ -^ -rt * Tt TJI Tt< o LI in LT: irt ut ut u^ i_t ir: u7 ir: L-: ITS i^ r: i^ ut ir: i^ o DO^jaO-4*C4COCD-MO)OeOO>t>aOOOOQOO^C4gCDG9 ^^^gggggggg^g^ggg^^^w^sSSSSSSg b Of^i^M(NCOCO^4|l^^cbcD^^a^CAaOQ>^f^C^(NCQCO-4*4HlQ ccc y ^fCcoccrcccc 7ccr i tfCCCfCccr7rcotrC *Tti * ^T3-TtTt TfTtTri <* ao t^ra 1-1 10 o cp eo d C4 co co *- co os e eo 10 go H; *- a oo] CD 04 o o oi i-lOOSQOOOt^t COCOCOCOtOCOt^t^OOaSO" iCi-nJCOt^OC^^It^OCOCO !M c: cx> t-~ ^ L-^ -- 77 01 o ~. cc t^ ^2 -t tr: Tt cc C<1 > o C ~. x t^ c^ o o i^ccccc5O"!>4rc^rL~^i^t-^xciO iC-jrC tut^r^xxc; O^c-irc (>l<^<^^^cc^rc?c^r^r^r^c^rc^^^^^^Tt < Tt T^ *^i- ri ; -:L :o b Lt O u-5 O & O LO O ^ O U2 O O O O O O O O Ut O >--: O tr: O O O O *.l* re u-t oo ^ l5c?|8Sc|c?|fS?222?S2?:2?85lSSS|l8|, 12 THE ELLIPTIC FUNCTIONS. 12. We notice that when the modular angle is 15, then log K lK =-2385606=1 log 3, so that K /K=^3: this will be proved subsequently ; but it shows here that the period of a pendulum oscillating through 300 is ^/3 times the period when the pendulum oscillates through 60. Again we shall prove subsequently that, if K /K=J t 7,iheiL2icK = l , so that equal parallel horizontal chords, BE the higher, and W the lower, each of length one-eighth the diameter, cut off arcs of the circle below them, which would be swung through by the pendulum in times which are in the ratio of ^/l to 1. Many other similar numerical examples can be constructed when the Theory of the Complex Multiplication of Elliptic Functions is studied. 13. When a=j7r, the pendulum drops from a horizontal position and swings through two right angles, as in the Navez Electro-Ballistic Pendulum ; and now K=K , and the modular angle is JTT. Table II. from Legendre s Fonctions Elliptiques, t. II., gives to five decimals the value of u = F<f> for every half degree in the value of <, when the modular angle is 45 ; and thence by means of the preceding formulas which determine the motion of the pendulum by elliptic functions, the pendulum can be graduated so as to measure small intervals of time A = Au/%, as required for electro-ballistic experiments. Then from Table II., when K = K , and K = K=$J2, en u = cos <, sn u = sin <, dn u = ^/(l J win 2 ^). 14. Generally in the pendulum, K=^nT, so that the period When /c = 0, K=\ir, and the period is ^^(l/g), as proved otherwise in the ordinary elementary treatises, for small oscillations of the pendulum^ But in the finite oscillations of the pendulum, with then ( 8) dO/dt = ln K en 4-Kt/T, ,sin-0= K Putting = 0, 16 = 0, we find cnO = l, snO = 0, dnO=l ; THE ELLIPTIC FUNCTIONS. 13 and putting t = J7 7 , u = K, = JTT, when the pendulum has swung to OB, en A" = cos JTT = O, sn K=l, dn K = K ; while putting t = \T, u = 2 A", when the pendulum is swinging backwards through the verti cal OA, cu2K=-l,sn2K=0, dn2A =l; analogous to the values of cos and sin 0, for $ = 0, ^TT, TT ; so that 2K is the Aa/ period of the elliptic functions, corre sponding to the half period TT of the circular functions. rir ( p rir /-$ Since /c?0. A< = Yd0/A0 /d<f>/&&lt;f> = 2K u, if = am u, 000 therefore am(21Ttt)= -jr<p= 7ramt; and generally am(2 m A" u) = m?r = m?r am ^6 ; so that cn(2??iJ*ri6) = COS(??ITT am IL) = ( 1 ) y cn u, sn(2mKu) = sin(m7rain u)=( l) m sn u, while dn(2m-fiT it) = dn u : analogous to cos(m?r 6)= ( l) m cos 0, sin(m7r#) = ( - l) ?n sin ; and representing the motion, ??i half periods, past or future. 15. The degenerate Circular and Hyperbolic Functions. As a increases from to TT, K increases from to 1, and K from |-TT to infinity; the pendulum has now, with /c = l, just sufficient velocity to carry it to the highest position, and this will take an infinite time. For with a = ?r, equation (3), page 3, becomes I (dO/dt) 2 = rr(l + cos 0) = 2n 2 cos 2 J0 ; = log tan JO + 6) = log(sec J0 + tan J0), which is infinite when O = TT. In Small oscillations the period is 27r/n, and the motion of M, the projection of P on the horizontal axis Ax, is then a Simple Harmonic Motion (s.H.M.) given by the differential d 2 x equation _ + n -x = 0, the solution of which is o; = .4cos^, or Bsmnt, or A cosnt+Bsmnt, or acos(7i^-f e) ; so that n is the constant angular velocity round D of the point Q on the infinitesimal circle AQD, as in the cycloid. 14 THE ELLIPTIC FUNCTIONS. In Kepler s Problem in Astronomy, n represents what is called the mean motion of a planet or satellite, and nt ornt+e the mean anomaly ; a satellite of Jupiter, when observed in the plane of its orbit, supposed circular, will appear to move with a s. H. M. But with *==!, putting J0 = = angle AEP (fig. 3) nt =ysec 0^0 = log(sec + tan 0), so that sec + tan <f> = e nt , sec tan = e~ w< , sec = \(e ni + e~ nt ) = cosh nt, tan = (e nt - e ~ nt ) = sinh nt, sin (f> = tanh nt, cos = sech nt, tanJ0 = tanhJ?i, and so on. Also dO/dt = 2n cos J0 = 2n sech nt ; so that if the angular velocity of the pendulum in the lowest position OA is 2n, the pendulum will just reach the highest position OE ; but the time occupied in reaching it will be in finite, since 6 = 7r, 0=.j7r makes nt and therefore t infinite. The velocity of P in any position is l(dO/dt) = 2nl cosW = n . EP, and therefore varies as EP. If EP in fig. 3 is produced to meet Ax in M t then AM = AE tan0 = 21 sinh nt, EM = EA secJ0 = 21 cosh nt ; so that, if AM or EM is denoted by x, d2x -n*x-0 ~dt* the general solution of which differential equation is x A cosh nt-\-B sinh nt. 16. When the pendulum just reaches the highest position OE, ic = l ; and u (or nt) and 0, the c.m. of the angle AEP, are connected by the relations u =y*sec d<j> = log (sec <f> + tan 0) = cosh ~ ^ec $ = sinh ~ Han <p = tanh - 1 sin = 2 tanh ~ : tan J0. Conversely = cos ~ 1 sech u = sin ~ a tanh u = tan ~ J sinh u = 2 tan ~ Hanh Ju ; and then is called by Professor Cayley the Gudermannian of u, and denoted by gdu; so that if = gdu, then u = gd " J = log (sec + tan 0) = cosh - ^ec 0, etc. THE ELLIPTIC FUNCTIONS. 15 Holiel proposes for <p the name of hyperbolic amplitude of 16, with the notation = amh u, instead of gd 16 ; so that amh u sec <>d 16=/S< o or ^ = arnh u = /sech udu = cos ~ 1 sech 16 = sin Hanh u, etc ; analogous in the general case of the elliptic functions, for any modulus /c, to ( 7) = / u, etc. As degenerate forms, when K = 1 , en it = sech u, sn 16 = tanh u, dn u = sech u ; while, with /c = 0, en u = cos 16, sn u = sin it, dn u = 1 . Thus, when /c = l, the elliptic functions degenerate into the hyperbolic functions ; and, when K = 0, into the circular func tions ; but with any other value of the modulus K, the elliptic functions must be considered as new functions, of a higher order of complexity than the circular or hyperbolic functions. The following Table, from Legendre, F. E., t. II., Table IV., gives the values of 16 = log (sec 4- tan </>) = log tan( JTT + J0) for every degree of radians ; whence the numerical values of the hyperbolic functions of u can be determined, by aid of a table of circular functions, and by the relations cosh u = sec 0, sinh u = tan <p, tanh u = sin 0, .... For values of u greater than about 4 the Table fails ; but then it is sufficient, to two decimals, to take cosh u = sinh u = Je M ; Iog 10 cosh 16 = Iog 10 sinh u = Mu log 2 ; or, to a closer approximation, Iog 10 cosh u = MIL log 2 -f Me ~ 2u , . . . , Iog 10 sinh u Mu log 2 Me ~ 2 ", . . . , Iog 10 tanhu= -2 Me~ 2u ..., M denoting the modulus Iog 10 e. (Proposed Tables of Hyperbolic Functions, Report to the British Association, 1888, by Prof. Alfred Lodge.) 16 THE ELLIPTIC FUNCTIONS. TABLE TIL u u M o-ooooo o-ooooo 30 0-52360 0-54931 60 1-04720 1-31696 1 2 3 0-01745 0-03491 0-05236 0-01745 0-03491 0-05238 31 32 33 0-54105 0-55851 0-57596 0-56956 0-59003 0-61073 61 62 63 1-06465 1-08210 1 -09956 1 -35240 1-38899 1-42679 4 5 6 0-06981 0-08727 0-10472 0-06987 0-08738 0-10491 34 35 36 0-59341 0-61087 0-62832 0-63166 0-65284 0-67428 64 65 66 1-11701 1-13446 1-15192 1-46591 1-50645 1-54S55 7 8 9 0-12217 0-13963 0-15708 0-12248 0-14008 0-15773 37 38 39 0-64577 0-66323 0-68068 0-69599 0-71799 0-74029 67 68 69 1-16937 1-18682 1 -20428 1-59232 1 -63794 1-68557 10 0-17453 0-17543 40 0-69813 0-76291 70 1-22173 1-73542 11 12 13 0-19199 0-20944 0-22689 0-19318 0-21099 0-22886 41 42 43 0-71558 0-73304 0-75049 0-78586 0-80917 0-83284 71 72 73 1-23918 25664 27409 1-78771 1-84273 1-90079 14 15 16 0-24435 0-26180 0-27925 0-24681 0-26484 0-28295 44 45 46 0-76794 0-78540 0-80285 0-85690 0-88137 0-90628 74 75 76 29154 30900 32645 1-96226 2-02759 2-09732 17 18 19 0-29671 0-31416 0-33161 0-30116 0-31946 0-33786 47 48 49 0-82030 0-83776 0-85521 0-93163 0-95747 0-98381 77 78 79 34390 36136 3788 1 2-17212 2-25280 2-34040 20 0-34907 0-35638 50 0-87266 1-01068 80 1-39626 2-43625 21 22 23 0-36652 0-38397 0-40143 0-37501 0-39377 0-41266 51 52 53 0-89012 0-90757 0-92502 1-03812 1-06616 1-09483 81 82 83 41372 43117 44862 2-54209 2-66031 2-79422 24 25 26 0-41888 0-43633 0-45379 0-43169 0-45088 0-47021 54 55 56 0-94248 0-95093 0-97738 1-12418 1-15423 1-18505 84 85 86 46608 48353 50098 2-94870 3-13130 3-35467 27 28 29 0-47124 0-48869 0-50615 0-48972 0-50939 0-52925 57 58 59 0-99484 1-01229 1-02974 1-21667 1-24916 1 -28257 87 88 89 51844 53589 1-55334 3-64253 4-04813 4-74135 30 0-52360 0-54931 60 1-74720 1-31696 90 1-57080 infinite. THE ELLIPTIC FUNCTIONS. 17 Considered. as a function of the latitude <, u was called the meridional part by Edward Wright, 1599, who first employed it for the accurate construction of the parallels of latitude on the Mercator Chart, by making the ratio of the distance from the equator of the parallel of latitude < to the distance between the meridians whose difference of longitude is equal to the ratio of u/<j> ( 98). 17. Returning to the general elliptic functions, we notice that en 2 u + sn 2 u = l, dn 2 u -f- /c 2 sn 2 w, = 1 , or, in a tabular form, en sn dn en u 7(1-811%) snu dnu whence any one of the three elliptic functions en, sn, dn, can be expressed in terms of any other ; the three functions are thus not absolutely necessary, but all three are retained and utilized for simplicity of expression, as sometimes one and sometimes another is most appropriate for the particular pro blem in hand ; in the same way, of the circular functions cos 9, sin 9, tan 0, cot 9, sec 9, cec 0, vers 9, one would be sufficient, but all are useful ; and so also with the hyperbolic functions cosh u, sinh u, tanh u, For the reciprocals and quotients of the elliptic functions en, sn, dn, a convenient notation has been invented by Dr. Glaisher, according to which 1/cn u is represented by nc u, 1/sn u by ns u, 1/dn u by nd u, en u/dn u by cd u, and so on. In this manner sn u/cn u would be denoted by sc u ; but it is more commonly denoted by tanam u, abbreviated to tn u ; while en u/sn u or cs u would be denoted by cotam u, or ctn u. , According to Clifford (Dynamic, p. 89) we might abbreviate the designation of the hyperbolic cosine, sine, and tangent to he, hs, and ht ; or we may write them ch, sh, th ; with en, sn, tn for the elliptic functions ; and merely c, s, t for the circular functions. G.E.F. 18 THE ELLIPTIC FUNCTIONS. 18. Pendulum performing complete revolutions. Secondly, suppose the pendulum performs complete revolu tions (fig. 3). We have seen previously ( 15) that if the pendulum has an angular velocity 2n = 2^/(g/l) in the lowest position, it will just reach the highest position; and therefore if this angular velocity is increased, the pendulum will perform com plete revolutions. The integration of equation (1) in the form or ^v 2 /g + ANAD, a constant, denoted by 2R, shows that the velocity of P is that which would be acquired in falling freely from the level of a certain horizontal line BDB , which now does not cut the circle, as in fig. 2 when the pendulum oscillated, but lies entirely above the circle, as in fig. 3, at a height 2R above the lowest point A ; and the im petus of the velocity of P is the depth of P below BB . Denoting the angle AEP by 0, so that = |$, then 2l 2 (d<p/dt) 2 = g(2R I vers 20) == 2g(R . /cZ0\ 2 g or l-^-J = - on putting /c 2 = l/R = AE/AD ; and n 2 =g/l, as before ; so that nt/ K =f(l - K 2 sin 2 0) - cZ0 = P(0, K\ in Legendre s notation ; and inverting the function according to Abel s suggestion, with Jacobi s notation, and now, with Gudermann s abbreviated notation, cos ^0 = en. nt/K, d0 ^n , ,, di = K /* AN= I vers = 2Z sin 2 = AEsn 2 nt/ K , NE AE cri*nt/K, ND = AD dn 2 nt/K, AP = AE sn nt/K, PE=AE en nt/ K , NP = 21 sin ^0 cos J$ = AE sn nt/K en nt/K. THE ELLIPTIC FUNCTIONS. 19 19. The time of moving from A to E is obtained by putting = IK, and is therefore Kic/n ; and therefore the period, or time of a complete revolution, is 2KK/n (not 4jfic/n). With the series for K as given in 11, and with K 2 the period of the pendulum for a complete revolution is The analogous expression for the period when the pendulum oscillates, rising on each side to a height 2R, less than 21, is, as in 11, 8 Putting /c = 1, and R = I, makes K infinite, and brings us back again to the separating case between oscillations and complete revolutions of the pendulum ; and we thus regain for this case the original expressions involving hyperbolic functions, previously investigated in 15. But as K now diminishes again from 1 to 0, the pendulum revolves faster and faster, until finally, when Ac = 0, we must suppose the pendulum to revolve with infinite angular velocity, the fluctuations of which for different positions of P are in sensible ; and the period is now zero. 20. We notice that, in the circle AQ (fig. 2) the point Q moves according to the law ^ = am nt, so that Q moves round in a circle, centre 0, in fig. 2 like the point P making complete revolutions in fig. 3. But now, in the motion of Q, gravity must be supposed diluted from g to K 4 g ; for if R or K-l denotes the radius of the circle A Q, g the diluted value of gravity, and n = *J(g /R) the speed of the pendulum CQ, then we must have $ = am nt = am rit/K, so that n = Kn We may dilute gravity in the circle A Q by inclining the plane of the circle to the vertical at an appropriate angle. 20 THE ELLIPTIC FUNCTIONS. 21. Another way of diluting gravity would be to replace the circle AQ by a fine tube in the form of a uniform helix with horizontal axis through its centre G perpendicular to the plane of the circle AQ, and to suppose the particle Q to move in this helix under gravity. Then we shall find that if the length of one complete turn of this helical tube is equal to the circumference of the circle AP, the particle Q moving with velocity due to the level of E will follow the motion of the particle P moving on the circle AP with velocity due to the level of D, so that PQ will always be horizontal, if once it is horizontal, and P, Q will always be at the same level during the motion. For in this case the mechanical similitude is secured by in creasing the square of the velocity of Q in the ratio of 1 to l//c 4 , instead of diluting gravity to /c 4 </. We may secure the same effect by supposing Q to be a point on a pendulum CQ , of length greater than GQ ; or else of length GQ, but of which the axis (7 is cut into a smooth screw of appropriate pitch ; or else engaging with teethed wheels, so as to increase the angular inertia about G. 22. If we produce GQ to any fixed distance CQ =l , then Q will also perform complete revolutions like a pendulum of length , with gravity changed in a certain fixed ratio depend ing on I ; and we can keep gravity unchanged by choosing I so that n 2 = g/l = K V = K 2 g/l, or I = l/ K * = lcosec 2 a; and now Q revolves with velocity due to a level at a height 2//c 4 = 2cosec 4 Ja above its lowest position; so that the period of revolution of a simple pendulum of length I cosecHa, when the velocity is due to the level of a line at a height 2cosec 4 Ja above its lowest point is equal to the time of oscillation of a simple pendulum of length I through an angle 2 from rest to rest. These problems on the pendulum have been developed here at some length, in accordance with the idea of this Treatise, that it is simple pendulum motion which affords the best concrete illustration of the Elliptic Functions. Similar principles are involved in the following three theorems, which the student can prove as an exercise in the manner employed for the cycloid in 10. THE ELLIPTIC FUNCTIONS. 21 1. If two vertical circles, of diameters AD and AE, touch at their lowest points A, the time of oscillation from rest to rest of a particle in the circle AE with velocity due to the level of D will be to the time of revolution of a particle in the circle AD with velocity due to the level of E in the ratio of AEio AD (fig. 2). 2. Two particles move, under gravity, in vertical circles. The one oscillates ; the other performs complete revolutions. Prove that if the height to which the velocity of the first is due bears to the diameter of the first circle the same ratio as the diameter of the second circle bears to the height to which the velocity in it is due (the heights being measured from the low est points of the circles) the ratio of the squares of the times in corresponding small arcs and therefore the squares of the whole times of oscillation and revolution will be that com pounded of either of the before-mentioned equal ratios and the ratio of the diameters of the circles. 3. Two equal smooth circles are fixed so as to touch the same horizontal plane, their planes being at different inclinations ; two small heavy beads are projected at the same instant along these circles from their lowest points, the velocity of each bead being that due to the height of the highest point of the other circle above the horizontal plane, show that during the motion the two beads will always be at equal heights above the hori zontal plane. 23. We have compared the motion of the pendulum in fig. 1 with that of the simple equivalent pendulum composed of the particle P moving on a smooth circle, or at the end of a fine thread or wire OP ; oscillating from B to E in fig. 2, and performing complete revolutions in fig. 3, the velocity of P at any point being that acquired in falling from the level of D. Taking as coordinate axes the horizontal and vertical axes Ax and Ay through A, and referring the motion of P to the coordinates x and y, then since P describes the circle AP of radius I, x 2 = 2ly y 2 . Denoting by v = ds/dt the velocity of P, then by the principle of energy i^ 2 /# = 2P y, 2R denoting the height of D above A. 22 THE ELLIPTIC FUNCTIONS. dx l But since -- _ dx 2 _ V *~ +2 ~- while (ds/dt) 2 = g(2R - y) ; so that l 2 (dy/dt) 2 =g(2R- y)(2ly - dt I called an elliptic integral in /, and of the ^rs^ kind. 24. Firstly, if the pendulum oscillates, R is less than I, and 2/ oscillates between and 2R ; and the integral is reduced to Legendre s canonical form by putting y = 2R sin 2 ; when nt =(! - K 2 sin 2 ) - 4d = ^(0, /c), where K 2 = R/l, n 2 =g/l , and therefore with Jacobi s and Gudermann s notation, and y = 2J? sn% = 2^/c 2 sn 2 ?i^ a; = 2fo sn ^ dn nt ; or ^=^Dsn 2 ^, ND = ADcn 2 nt, NE=AjEdu 2 nt, as before, in 8. 25. When /c = 0, the oscillations are indefinitely small ; and now y = 2^ sin 2 ?i, where R is a very small quantity ; and nt = I - - vr = sni ~ 1 A/ Jt - ^2R an ordinary circular integral. It was Abel who pointed out (about 1823) that in looking only at the Elliptic Integrals, mathematicians had been taking the same difficult point of view as if they had begun to deduce the theorems of elementary Trigonometry from an examination of the properties of the inverse circular functions, as deduced from the circular integrals. (Niels-Henrik Abel. Tableau de sa vie et de son action scientifique. Par C. A. Bjerknes. 1885.) THE ELLIPTIC FUNCTIONS. 23 26. Secondly, if the pendulum performs complete revolu tions, as in fig. 3, R is greater than I, and y oscillates in value between and 21 ; we now reduce the elliptic integral in 23 to Legendre s standard form by putting y = 2l sin 2 0, when nt/K=f(lK 2 sin 2 0)"W^=^[^, K) where K 2 = l/R } the reciprocal of its former expression ; and now = &m(nt/K, K), y = 2l sn?nt/K, x = 2lsii nt/K en nt/K ; or AN=AEsu 2 nt/ K , NE=AEcu 2 nt/ K , ND = AD dn*nt/ K , as proved before, in 18. 27. In the separating case between oscillations and complete revolutions, R = l, and now K = 1 ; and y = 21 sm 2 = I vers2< = I vers ; also ( 23) nt =ysec Qd^ = log(sec <p + tan 0) = cosh " 1 sec = sinh ~ Han = tanh ~ x sin = 2 tanh ~ Han J< ; so that < = gd nt, or arnh nt, and sec = cosh nt, tan = sinh nt, sin = tanh nt, y = 21 tanh 2 ?i, x = 21 sech nt tanh nt, as before, in 15. 28. Landen s Point. With centre E in fig. 2 and radius j5 describe a circle cutting the vertical AE in L ; then Z is an important point in the theory of pendulum motion and elliptic functions, called Landen s point. Since EB* = ED.EA= EC 2 - CA\ therefore the circle, centre E and radius EB, will cut the circle AQD, centre C, at right angles ; and since LC 2 +CQ 2 = LC- 2 +EC 2 -EL*=,2LC . EC, and EL = EB = 21 K , EC=l(l+ K ^, LC=l(l- K J. Now, by 20, the velocity of Q = J(2g . EN) = J(2g K * . ^Y) = n^(2l . EN) =n.LQ(I+ K ). Similarly in fig. 3, where P makes complete revolutions, the velocity of P = n.LP(I+K)/K, where the Landen point L is obtained by drawing a circle with centre D, cutting the circle ^.^orthogonally, and the vertical AD in L. 24 THE ELLIPTIC FUNCTIONS. We shall prove subsequently that any straight line through L divides the circle APE in fig. 3 (or the circle AQD in fig. 2) into two parts, each described in half the period. 29. Change from one modulus to its reciprocal. It is important for the simplicity and for convenience of tabulation of the elliptic functions that the modulus K should not exceed unity ; but the preceding reductions of the motion of the pendulum to elliptic functions, in the two cases in which the pendulum oscillates and performs complete revolutions, show us how to make the elliptic functions to a modulus /c, which is greater than unity, depend on the elliptic functions to the reciprocal modulus l//c, which is less than unity. For, on comparing the two expressions for y, according as the pendulum oscillates or performs complete revolutions, y = 2Rsn\nt, K ), or 2lstf( K nt, l//c), where /c 2 = Rjl ; so that K 2 sri z (nt, K) = sn 2 (V?i, I/K) ; or, putting nt = u, K sn(u, K) = sn (KU, I/K), so that dn(u, /c) = en (KU, l//c), cn(^, /c) = dn (KU, I/K). Independently, if we suppose < = am(^, /c), and if we put K sin (j> sin i^, then K cos <pd<j> = cos \/r d\js, and cos so that u = /(I /c 2 sin 2 </>)~*<i0 /s u /s KU = sec or and since /c sin = sin i/r, etc., therefore K sn(w., /c) = SV.(KU } l//c), etc. When u = K, = j7r, and i/r^sin" 1 ^; so that, if K is less /sin" 1 * ( 1 /c ~ 2 sin 2 \/r THE ELLIPTIC FUNCTIONS. 25 30. Rectilinear Oscillations expi^essed by Elliptic Functions. In simple pendulum motion, referred to horizontal and ver tical axes Ax, Ay, drawn through the lowest point A, we have shown in 24, 26, that y = 2lK 2 su 2 nt, x = 2//c sn nt dn nt ; or y = 2lsn?nt/K f x = 2lsn nt/K en ntJK ; according as the pendulum oscillates or performs complete revolutions. Treating the vertical motions separately, and differentiating according to the rules established in 7, we find, on taking y = 2/c 2 sn 2 ?i, dy/dt = 4ilnK 2 sii nt en nt dn nt d 2 y/dt 2 = 4<ln 2 K\cn?nt dn-nt - sn 2 nt dn 2 ?? t - K 2 su 2 nt ctfnt) --+ , by 17. I IK." 46-/C-/ Taking y = 2lsn 2 nt/K, we find in a similar manner both immediately obtainable from the equation of 23, whence I 2 (d 2 y/dt 2 ) = g(Rl -Ry~ly+ f^ We shall find similar expressions for d-y/dt 2 when y varies as cn 2 ?i or dn. 2 nt, all of the form Let us determine then, as exercises in the differentiation of the elliptic functions, the acceleration d 2 x/dt 2 , and thence the force at a distance x t which will make a body oscillate in a straight line according to one of the laws x = a en nt, sn nt, dn nt, tn nt, nc nt, ns nt, .... Taking x = a en nt, dx/dt na sn nt dn 7i d 2 x/dt 2 ri 2 a(cn nt dn 2 nt K 2 su 2 nt en nt) Ml 26 THE ELLIPTIC FUNCTIONS. so that jp reducing to zero when /c = 0. It is often simpler to find dx/dt, and then to express as a function of x ; and then a differentiation with respect to t will give d 2 x/dt 2 immediately as a function of x. Thus, if x = a sn nt, dx/dt = na en ?i dn nt 80 that reducing to zero, when /c = 0. Similarly, if # = a dn 71*, Generally, when cc varies also as tn ?i, nc TI, . . . , we shall find a relation of the form which, when multiplied by dx/dt and integrated, gives KdxJMf = or dx/dt = an elliptic integral, of which the different expressions are given in Chapter II. 31. A Special Minimum Surface. Another interesting exercise in the differentiation of elliptic functions is to verify that the surface discovered by Schwarz (Gesammelte Mathematische Abhandlungen, vol. I., p. 77), cnx+cny + cnz + cnx cuy en = 0, with the modulus /c = J, is a minimum surface, having zero curvature at every point, and therefore satisfying the condition p, q, r, s, t having their usual meaning as partial differential coefficients of s with respect to x and y. THE ELLIPTIC FUNCTIONS. 27 Schwarz shows that this condition is equivalent to p v p. 2 denoting the principal radii of curvature of the surface (C. Smith, Solid Geometry, 255), where Y- P v = q J(p 2 + q 2 + lY J(f+f+l) Let us write c v s v d v for en x, sn x, dux-, and c 2 , s. 2 , d 2 , C 3 , S 3 , d 3 for the same functions of y and z. Then c x + c. 2 + c 3 + c^Cg = ; and differentiating with respect to x, But (l+c lC2 ) 2 so that s s (l + CiC 2 ) = s^, etc. ; _vA = __?A = r -,-. By symmetry, g= ___ so that we may write where D = (c^ 8l ) S By symmetry o 2 3 p, y p, T r- so that -t-- =0, provided that 28 THE ELLIPTIC FUNCTIONS. or or, since s x 2 = 1 - c^, d* = i(3 + q 2 ), or (<?! + c 2 + c s + c^) (3 - c 2 c 3 - c 3 c x - c^) = 0, and this is true, in consequence of the original relation Ci-f-Cg+Ca+CiCjCjrsO. The other relation 3 c 2 c 3 c^ c x c 2 = represents isolated conjugate points, where C 1 = C 2 = C 3 = 1 - Another minimum surface is tn y tn 2 + tn s tn 03 + tn x tn ^/ + 3 = 0, With K 32. Elliptic Function Solution of Euler s Equations of Motion. Before leaving the mechanical interpretation of elliptic functions, we may just mention here an important application, the application to the solution of Euler s equations of motion, for a body under no forces, moving about its centre of gravity, or about any fixed point. Euler s equations for p, q y r, the component angular velocities about the principal axes, are (Routh, Rigid Dynamics) where A, B, C denote the moments of inertia about the princi pal axes ; and two first integrals of these equations are = T, a constant ; =G 2 , a constant, obtained by multiplying Euler s equations respectively by (i.) p, q, r, and adding, (ii.) by Ap, Bq, Or, and adding ; and then integrating. Comparing these equations with the equations of 7, cn w = sn- u dn u, sn u = en u dn u, dn u = /c 2 sn u en u, where accents denote differentiation with respect to u, we notice that if A > B > C, and the polhode includes the axis (7, so that AT>BT>G 2 >CT, we may put u = nt, and THE ELLIPTIC FUNCTIONS. 29 p = P cn u, q = Q sn u, r = RdnU , and then, on substituting in Euler s equations of motion, B-G_nP A-C_nQ A-B_ K 2 nR A ~QR B ~RP C = PQ Putting t = 0, and therefore p = P, q = 0, r = R; then AP* + CR 2 = T, A 2 P 2 + C 2 R 2 = G\ G*-CT AT-G 2 so that P ~ ,., n , and then L B B-C~B(B-CY while n 2 = R 2 and AB ABC P 2 A A-B_G*-CTA-B & c B-C~AT-G 2 B-C If the polhode encloses the axis of greatest moment A, so -that AT > G 2 > BT > CT, we must put . 2) = P dn u, q = Q sn u, r = Rci].u; and then determine P, Q, R, n, K as before ; when AT-G 2 B-C 2 _- 9 _ ABC ~G 2 -CTA-B In the separating case, when G 2 = BT, then /c = l, and p = P sech nt, q = Q tanh nt , r = R sech nt ; so that, when t = 0, 2 _ G 2 B-C _ . J _G 2 A-B P ~ABA-C q ~ ~BCA-C and initially or finally, when t = + yz , 2 ) = 0, q= G ! B, r = 0; and the body is spinning about its mean axis B. But when the body is spinning about the axis of greatest or least moment, G 2 = A T= A 2 p 2 , or G 2 = CT= C 2 r 2 , and K = ; and the period of a small oscillation is 2-Tr/n, where . (A-B)(A-O (A-B)(A-0) ABC. BC P _ We shall return subsequently to these equations in Chap. III. CHAPTER II. THE ELLIPTIC INTEGRALS (OF THE FIRST KIND). 33. In Chapter I. we have immediately made use of Abel s valuable idea of the Inversion of the Elliptic Integral, which is the foundation of the modern theory of the Elliptic Func tions ; and we have considered the functions which are inverse to the elliptic integral, and treated them as the direct funda mental functions of our Theory. Previously to Abel s discovery (1823) it was the elliptic integral which was studied, as in the writings of Euler and Legendre ; and, in fact, in a physical and dynamical problem it is the elliptic integral which arises in the course of the work ; for instance in the form of the Equation of Energy, $(dx/dt) z = X, so that V 2 t=/dx/JX-, and now, when X is a cubic or quartic function of x, so that d 2 x/dt 2 is a quadratic or cubic, as in 30, the integral is called an elliptic integral of the first kind ; and we have to follow Abel and determine the elliptic function which expresses x as a function of t. To accomplish this, it will be useful to employ the notation of the inverse functions, given by Clifford (Proc. London Math. Society, vol. vii., p. 29 ; Mathematical Papers, p. 207) analogous to those used in Trigonometry for the inverse circular functions ; and to make a collection of all the important cases that can occur. 34. The Circular and Hyperbolic Integrals. Starting with the circular functions, sin x, cos x, tan x, cot x } ... , we have, in the ordinary notation, 30 THE ELLIPTIC INTEGRALS. 31 Cx fa I =sin- 1 ^ = cos- 1 /v /(l-^ 2 X */ ^ \J- ~*~ x j r\ r 7~ / aa " =cos- 1 ^ = sin- 1 x /(l-a; 2 ), +s ^/^-L ~~ X J x f X x dx 1 ^-j =cot-^ = tan--,etc. X We can employ a similar notation with the hyperbolic func tions, cosh x, sinh x, tanh x, coth #, . . . , and write X Jrf - = x-\-\ etc.; and the analogy with the circular functions is now complete, and the results can be more easily remembered and written down, than when the logarithmic function alone is employed. To avoid complications due to the multiplicity of the values of these and subsequent integrals, in consequence of the variable x assuming complex values and performing circuits of contours round the poles of the integral, we suppose for the present that x is real, and increases or diminishes continually, so as to assume all real values once only between the limits of integration; also that the positive sign is taken with the radical under the sign of integration ; we thus obtain what is called the principal value of the integral or inverse function. 35. The Elliptic Integrals. With the elliptic functions, snu, cnu, dnw, we have ( 7) d SD.U dcnu ddnu -j = en u dn u, -. = sn u dn u, j = /c 2 sn u en u : du du du 32 THE ELLIPTIC INTEGRALS. and cn 2 i> = 1 sn 2 u, dn z u = 1 /c 2 sn 2 w, ; so that, if x = snu, then en u = ^/(l a? 2 ), / ^^ 7(i-^.i- K vr u==sn " la; or sn 1(x> K) - ....... (1: when the modulus K is required to be put in evidence. Putting x=l makes the integral equal to K, the quarter period corresponding to the modulus K ( 11). Similarly, with x = en u, then sn u = */( 1 x 2 ), dn u A = -sn u nu = - /i ^^ y (1 -^. K HA*r w = cn la or so that the integral is K when the lower limit is 0. Again, with x = dn u t then >c sn u = ^/(l x 2 ), K en u = ^/(a 2 Ac 2 ) ; and ~=- K *suucnu=-J(l-x*.x*-K 2 ) ) du We may also put x = iuu, using Gudermann s abbreviation of tn u for tan am u ; and now ,_- J. 1 / \ / A \ and the integral is K when the upper limit is oo . Putting & = sin0, cos 0, A</>, or tan< in (1), (2), (3), or (4), reduces the integral to /(I /c 2 sin 2 0) ~ -cZ0 = u = F(0, /c) o so that = am u, and cos = en u, sin = sn u, A0 = dn u, tan < = tn u. THE ELLIPTIC INTEGRALS. 33 36. Thus, with a>b>x, x dx 1 .fx b " (5 indicating that we must put x b sin ; and then the integral is reduced to a a V a o Similarly, with oc > x > a, f x dx _1 Ja b\ , J (tf-a?.x*-}? a 81 WJ" indicating the substitution x a cosec (or a cec 0, as Dr. Glaisher writes it). Thus, for instance, with oo>o?>l//c, f^_ dx VJL \ J x/(! ~ ^ 2 1 ~ * 2 z 2 ) ~ W V Again, ; 37. As numerical examples, /i r / r -^ the integration required in the rectification of the lemniscate r >2 = a 2 cos20; so that r = a cn(^/2 s/a, J with Dr. Glaisher s notation ( 17) of new for 1/cnu. G.E.F. C / 34 THE ELLIPTIC INTEGRALS. Consider also the vibrations given by the dynamical equation d 2 x/dt 2 = 2n 2 x(c 2 x 2 ), as in 30; so that x = gives the point of stable equilibrium, and x = c gives the points of unstable equilibrium. Integrating, supposing the motion to start from rest where x = b, i (dx/dt) 2 = C- n 2 c (i.) When b 2 <c 2 , the motion is at the outset towards the origin, and dx/dt = n^/(a 2 x 2 .b 2 x 2 ), writing a 2 for 2c 2 b 2 ; so that P dx f dx _f x dx n Jj(a 2 -x 2 .b 2 -x 2 )-J JXj JX x = -\K sn~V ), with modulus , by (5) ; a\ b/ a J or x = bsu(Kant). (ii.) When b 2 = c 2 , dx/dt = n(b 2 - x 2 ) ; and, by 34, the ultimate state of motion is given by x = b tanh bnt, or b coth but, according as the motion falls away from the position of unstable equilibrium, towards or away from the origin. (iii.) When c 2 < 6 2 < 2c 2 , dx/dt = r* dx r M dx _ J x 2 -a 2 .x 2 -b 2 )~J X J(x 2 -a 2 .x 2 -b 2 ) b b (iv.) When b 2 = 2c 2 , nt = / ,, 2 pr = ,- sec ~ V> 6 or x = b sec bnt. (v.) When 6 2 > 2c 2 , we must write a 2 for 6 2 - 2c 2 ; and now dx/dt = + nj (a 2 + x 2 .x 2 - b 2 }, dx _ 1 J6 a ~v/( 2 + 6 2 ) CI] U /v /(^ 2 + or x = 6/cn /v /(a 2 + b 2 )nt =b nc^(a 2 + b 2 )nt. THE ELLIPTIC INTEGRALS. 35 38. So far the function X has been treated as an even quartic function of x, or as a quadratic function of a; 2 , resolved into two real factors ; but according to Prof. Felix Klein there are certain advantages in considering the integrals obtained by writing x 2 =z, in (1), (2), (3) ; and then, waiting k for * 2 , f J or 2cn-V(l-s), or 2dn- 1 v /( 1 -^) ............. C 11 ) Conversely, by writing for z the values & 2 , 1 # 2 , 1 kx 2 , we reproduce the integrals (1), (2), (3) from (11), by the simplest quadric transformations; and it will not cause confusion if we sometimes call k the modulus. For these and various other reasons, Prof. Klein suggests (Math. Ann. XIV., p. 116) that we should consider (11) as a more canonical form of the elliptic integral than (1), the form with which Legendre and Jacobi have worked. 39. Now, with X = x a.x {3.x y, and a>/3>y, we have, if GO > x > a, f J q-y , (12) indicating that we must put x y = (a y)cec 2 0, x a = (a y)cot 2 0, and then x - /3 = (/3 - y) A 2 cec 2 0, to reduce the integral to Legendre s canonical form =(! - k sin 2 o Similarly, by putting x - a = (a - /3) tan 2 0, x - /3 = (a - / x a-y.o;-/3 where Jl/ is used throughout to denote J^/(a y). .(13) 36 THE ELLIPTIC INTEGRALS. Thus, with -ao>x>l/k, integral (11) becomes f j 9 1 L -" SI _ J(x.l-x.I-kx) l/fc 1 I l-k , . ll-k.x - 1 - -=2dn- 1 A/- J~. .x 1 v cc 1 40. When a>#>/3, Z is negative, and y _ la-.x- - (14) /-J/fe J *J(- = cn -i /toir:* = dn -i J^=y ...... (15); \a jS.aJ y \cc-y and now the modulus K is given by K 2 = k f = (a /3)/(a y) t and the modulus is therefore complementary to the modulus in (12) and (13) ; and the form of the result in these and other subsequent integrals indicates the substitution required to reduce the integral to Legendre s standard form ( 4) ; while the results can be verified by differentiation. Thus, with l//c>&&gt;l, integral (11) is imaginary and may be written dx I -fee - f* dx _ l I x-l JJ(x.l-x.I-kx)~ yi-k.x .kfaB-t^^-llfc-^ mod. *T; i denoting *J( 1). THE ELLIPTIC INTEGRALS. 37 41. When (3>x> y, X is again positive, and P Mdx la-y.fi-x 7T -^p-y.a-x -m^JyptelZ.-to-iJsze. . ..(16) V/3-y.a-x \a 05 / 3/tZa /a-y ^ = sn - V^ x ..................... (17) with fc = 08-y)/( a -y), as in (12) and (13). Thus /- 1 do? ll-x j _ v or * / _ J J(x.l-x.\-kc) \1-A while the result is as in (11) when the lower limit is 0. 42. When y>x> oc,JTis negative, and =cn x Mdx ja-y f. ^r = sn- 1 A /- - L J(-X) \a-x with modulus k = (a /3)/(a y), as in (14) and (15). Thus, with 0>x> oc , integral (11) becomes / J* . _ ---- = 2idn~ 1 / */- --- j ^ o- / ! .i-a:.i- = sn Vr- a 38 THE ELLIPTIC INTEGRALS. 43. We notice that the substitution _ ,r --- , UI - ,r --- , ^ 75 ---- , - , # y p y & y p y # y a y makes dx y dy x y or changes (12) into (17), or (13) into (16). Thus dy r dx . J(x-a.x-/3.x- 7 ) j(y-a.y-/3.y- 7 ) a 7 where K 2 = k = (/3 y)/(a y). Again the substitution ., or = or = a-/3 a-2/ a-/3 a~2/ a-y a-y changes (14) into (19), or (15) into (18) ; and shows that dx fy dy 2A" where k = K 2 = (a- ft) /(a - y). The substitution which changes any one integral into another is obvious by inspection of the preceding results. 44. Thus the integral fdxj^/X can be written down, ex pressed by inverse elliptic functions, when X is a cubic form in x, resolved into its three real linear factors. For example, with a 2 > 6 2 > c 2 , d\ 2 J c 2 +X \ an integral occurring in the mathematical theories of Electricity, Magnetism, and Hydrodynamics, in connexion with ellipsoids. As another example, the student may prove that r _dS_ 47ra6c Jc_ /a 2 -6 2 \ J (xjaY + (y~lb) z + (z l c)* ~ J(a 2 - c 2 ) C W V a 2 - c 2 7 when the integration is extended over the surface S of the sphere x* + y* + z 2 = r* (W. Burnside, Math. Tripos, 1881). THE ELLIPTIC INTEGRALS. 39 45. When two of the roots, /3 and y suppose, of the cubic X = are complex, we combine (x ($)(x y) into the real quadratic (#-m) 2 + n 2 , suppose ; so that X = x a . (x m) 2 -f n 2 - Now we substitute X = (a;-m) 2 +7i 2 y ~(x-af~ x-a a quadric substitution, the #rap/i of which is a hyperbola, and find the turning values of ?/, say y 1 and ?/ 3 , the values of y which make the quadratic in x, have equal roots ; so that y : and y 3 are the roots of = 0, or ^ + ( m - a )y -, - B Then y - yi = -, y-?,= . (Zy_(a?-a? 1 Xa?-g 8 ). anQ 7 \9 5 aa? (OB a) 2 x l and 3 denoting the values of x corresponding to y l and 2/3, and therefore denoting the roots of the quadratic equation x 2 - 2ax 4- 2am - m 2 - n 2 = ; so that x = m + x = m Then / J _r ^ by (12), with k / = yJ(y l y^), k= J \ /* t/ 1/ \t/ 1 t/ o/ since 2/ : is positive and y z negative, or y l > y > > y y Again, with the same substitution, dx r dy cn-i /ft-y 2/3) >2/i-2/ -^ rcn- 1 ^^ ...... (23) by (19), to a modulus k the complementary modulus of (22), namely k = yj(y^ - y s ). 40 THE ELLIPTIC INTEGRALS. 46. We denote (a-m) z + n* by H\ and then and by means of the same substitution as in 45, dx en , (24); dx I JH--(a-x) ,\ ~v# CI iff+to^sy*) r , (25); indicating that the substitutions x a or a x = H(^ reduce the integrals to Legendre s standard form ; also that Thus, as numerical examples, /CO fJ ] / "I /O \ tit*/ JL - 1 / ^ ^ ^v \ i /" g da; J_ y x/a-^ 3 )"^ with 2/c/ = J = sin 30, K = sin 15, K = sin 75. 47. We notice that ^ = ITT when x = aH ) so that " /* ^ y /v /{ a; --(-W ) 2 +^ 2 } +H = /: a+ ^L _J?L_ Jj{x-a.(x mf+n? a x . (x m) 2 + n 2 } (27) -00 /OO xV/v, 3 ,,= / 1 / 7**^ 1 \ \/3+l ix r^^ dx THE ELLIPTIC INTEGRALS dx /* dx r x/a-^)y - -v/3+l But, by the Cubic substitution x = (4< so that or ^(sin 75) = ^/3-F(sin 15), that is, K IK= ^/3, if /c = sin 15, as stated in 12. 48. Degenerate Elliptic Integrals. When the middle root /3 of th,e cubic X = approaches to coincidence with either of the extreme roots, a or y, or when the pair of imaginary roots become equal, the elliptic integrals degenerate into circular or hyperbolic integrals. j> We notice, from 16, that when & = 0, sn -1 ^ becomes sin" 1 ^, cn 1 ^ becomes cos" 1 ^, etc.; and that, when k = l,sn.- l x becomes tanh 1 ^, cn 1 ^ or dn- 1 ^ becomes sech" 1 ^, and tn- 1 ^ becomes sinh ~ l x. Thus, when k = l, the integral (11) /dx _ _ r dx /(x.l-x. i-kx) J(l-x)Jx = 2 tanh- = 2 sech~ __. 1 a; \1-OJ a? This supposes that x < 1 ; but with oo > x > 1, ^^ oo 7 y (!)> = 2 coth " V^ = ^ cosech- V(s- 1) = 2 sinh- 1 J - = 2 cosh-i/ = sinh -i V^. \ic-l \-l ic-1 But when & = 0, the integral (11) becomes . = 2sm~ 1 /v /ic - 2 cos- r 1 dx _ _j x = 2sin- 1 v /(l X) = TT sm~ l 2^(x. Ix). 42 THE ELLIPTIC INTEGRALS. 49. Making /3 = y, or a, in the integrals (12) to (19), and still denoting J*/(a y) by M, then (i.) with oo>aj>a, Mdx Ix-a la y / =Sin~ 1 A / --- ^ VOJ- VoJ = COS - y 1 i*/(a y & a) - 1 v -flfcfcc . ! a-a sm = cos , /a-y " 1 ,*/ -- ^-; A aj-y T 60 Mdx , . /a-y . , , /a-y /7 -- - -- ^ = - --- ^- = - 1 -- * J (x-a this integral being infinite when a; = a. (ii.) With a > a; > y, f* Mdx . Ix-y /a-y /7 -- ~ -- - --- ^- -- * y(a- 7 which is infinite when x = a ; f a Mdx . la-x /7~ -- \~~77 - r = smh~ 1 A /- -^ J (x-y)J(a-x) \iC- which is infinite when a? = y. (iii.) With y > cc > oo , Mdx . , /y-aj /-y ~ ~ = 1 - ---- = " 1 -- L a? a-x , = cosh - J A / = sinh - \/ "a an Vy a? \v x this last integral being infinite when cc = y. The limits have been chosen so as to exclude these infinite values. 50. Weierstrass s Elliptic Functions defined. When the general cubic expression X is given, not resolved into factors, then Weierstrass s notation becomes useful, and may be defined here. THE ELLIPTIC INTEGRALS. 43 Weierstrass writes s+f for x, and chooses / so as to make s 2 disappear in the new value of X, which he denotes by J5 ; and thus S = 4s 3 g 2 s <? 3 , where g 2 and g 3 are called the invariants ; so that the integral dx ds x ds u suppose f J and now, inverting the function in Abel s manner, s is an elliptic function of u, denoted by pit- in Weierstrass s notation, so that =rl8i or p-(; g v g s ) ............. (A) when the invariants g 2 and (/ 3 are to be put in evidence. 51. In Weierstrass s notation we are independent of the particular resolution of S into factors ; but by what precedes in equation (12), if, when $ is resolved into real factors, S = 4(s 6^(8 e. 2 )(s e B ), with e l > e. 2 > e 3 , then, with x > 6> > e v C ds 1 _^ _ l /e 1 ~<? 3 e)^ V - s e l 1 , , !* e< = vdn~ by (12); so that (B) The value of u for s = e l is denoted by o^, and called the real half period; and by (20) we notice that Wl =f x ^=r^ s =^--- (28) e x ^ e 3 andby(13)and(B),/ %< -^ < ==p- 1 ( ei ~ gg ei ~ g8 + g 1 ) (29) w/ v ^ ^ s e i With e 2 >s>e 3 , ^jS is again real, and by (16), (17), and (B), 44 THE ELLIPTIC INTEGRALS. 52. For values of s between e 1 and e 2 , or between e z and ~ , *J& is imaginary ; however, the value of Jdsj \JS be tween the limits e z and oo is denoted by o> 3 , and called the imaginary half period ; so that, by (21), /-i ds r<* ds _ iK f ^ = J jsy X/S T^S) ............. (82) 2 and, from (12) and (14), * 2 = ( 2 - 8 )/(l ~ *3)> K* = ( e i ~ 2 )/(l ~ S )- Also, from (14) and (15), with e 1 > s> e 2 , / c i ds . 1/01 e^e, o \ ^s =tp ( - t-s <i; ^ ~4 ......... (33) / s <s . ,/e, e 2 .e 2 60 \ ^=^-v ;_; -*-<*> * -*)j ....... and, from (18) and (19), with e s > s > oo , / e a ds /S fj Q -Ta^P H-*; 9 -9 3 ) (36) V^ 53. The quantity gf 2 7# 3 2 is called the discriminant, and is denoted by A ; it is called the discriminant, because the roots of 8 = are all three real, or one real and two imaginary, according as A is positive or negative ; and A = 0, when two roots are equal. Since S = 4 3 - g 2 s -g 3 = 4(s - e^s - e 2 )(s - e^\ therefore e^e^e^O, and g 2 = - 4(e 2 e s + e^ + e^ = 2(e* + e 2 2 + Therefore and This quantity 5f 2 3 /A is called by Klein the absolute invariant, and denoted by J ; and then, with k for /c 2 , - ,_ _- r ~ 2 2 THE ELLIPTIC INTEGRALS. 45 54. For the present we reserve the difficulties of interpreta tion of the multiple values of the integral u =fds/*JS, due to s being allowed to assume complex values, and to perform circuits round the poles, branch points, or critical points, so called, of the integral, given by the roots of S = 0. We suppose the variable s to pass once through all real values from oo to oo ; and now (i.) oo > s > e, / or u=v l -dslJS=< a ,- 9 --*-*+e, ........ (37) which, employing the direct functions, expresses the relation *->-*-*=$? ................... (38> (ii.) e l >s>e 2 , (39) or u = CD, + a), -e 2 ; g r -g s ......... (40) (iii.) e 2 > s> e 3 , u = w + w (41) or u = 20)! -f- c 3 /ds/^/S < 3 (iv.) e 3 > s > co , -jr.); (43) / 46 THE ELLIPTIC INTEGRALS. or u = 2^! + 2ft> 3 /ds/^/8 00 Thus /ds/^S=2a) l + 2co 3 , (45) and 2ft)! is called the real period, and 2w 3 the imaginary period of Weierstrass s elliptic function pu. With Argand s geometrical representation of a complex quantity, such as x + iy, the complex quantity u = ta> l + 1f( l 6 3 (0<t<l,0<t <l) represents all points lying inside a rectangle, called the period parallelogram. As 8 or $u diminishes continually from oo to GO , the argu ment u describes the contour of this rectangle ; and for (iii.) t^ + a> 3 (I>t> 0), (iv.) t a> s (l>if> 0), the values of s or $>u- are real, and lie in the intervals (i.) ao>s>e lt (ii.) e 1 >s>e 2 , (iii.) e 2 >s>e 3 , (iv.) e 3 >s> CXD ; while the corresponding values of $ u are taken as (i.) negative, (ii.) positive imaginary, (iii.) positive, (iv.) negative imaginary. For any point u inside the rectangle $u assumes a complex value. (Schwarz, Elliptische Functionen, p. 74.) 55. In the same way, with the integral (11), denoting its value between the limits oo and z by u, (i.) oo >z>l/& (39), = 2K-2sn-^f^\ ................... (46) (ii.) l/k > z > 1 ( 40), (47) (iii.) 1>0>0(41), - kz .................................. ( 48 ) THE ELLIPTIC INTEGRALS. 47 (iv.) 0>2?>-oc (42), (49) Therefore f~ r =-^- 7 ^ = 4A r +4iJT, .............. (50) ^/ *J(Z . 1 2? . 1 A 1 ) and 4JT and 4^Ar are called the real and imaginary periods of the corresponding elliptic function, in this case sn 2 |i,. 56. But if we take Legendre s and Jacobi s fundamental integral Jdxj ^/X, where X = 1 x 2 . 1 K 2 x 2 , and denote r Idxj ^/X by u, then, by the preceding article, with x~ for 0, (i.) CC>X>\JK, 1 / ^2^2 _1 " KV& ~~ -L / K -i \ (ol) (ii.) I " I - /C -, K ^ ........................ (52) (iii.) 1>*> 1, = J- /C (iv.) -1> a; >-!/*, /c 2i^-^sn- 1 / - 1 , K .................... (54) (v.) -l//t ..................................... (55) Therefore *,(I-.x*.I-K*a?)-*dx = 4K+2iK -, .............. (56) -00 and 4/f and 2iK are called the periods of the elliptic func tion sn u. 48 THE ELLIPTIC INTEGRALS. 57. If, with l>x>-l, and X=I-x 2 .I-A z , we denote r* r i the iategr&lfdx/JX by u ; then/dfe/^/Z=JT ( 11); and ( 41) or, employing the direct functions, / T sn and then ( 17) \ 1X 2 - w ) == or cdw; ......... (57) (58) < 59 > relations analogous to equation (38) ; or to the relations sin( JTT 0) = cos 0, cos( JTT 6) = sin 9, of the circular functions of Trigonometry. 58. When the discriminant A of 8 is negative, and two of the roots of the equation 8 = are imaginary, we take e 2 as the real root, and combine the product s e^.s e^ into (s m) 2 +7i 2 , as in 45 ; and since therefore m = Je 2 , # 2 = 3e 2 2 47i 2 , # 3 = while H 2 = (6 2 - m) 2 + ?i 2 = f e* + ?^ 2 , 4 A 2 = n*/H 2 = 47i 2 /(9^ 2 2 1 - 16/cV 2 = 3^ 2 /(9e 2 2 + 4 2 ), A = 3 , fha r_ 2 _ _ = " _ 108A 2 - 2 59. Now, as in 45, by means of the quadric substitution, ,2 ,....-. (60) THE ELLIPTIC INTEGRALS. 49 da- (s-e,) 2 -# 2 (s-s)( s -s) ds= (s-e z f (A) 2 (s-^) 2 (s-s 3 ) 2 while oe^-, o--^ = _-__, provided S 1 = Thence s l + s 3 = 2e 2 = J( l + e 3 ) - e 2 - e 2 = - f 6 2 ~ e z 5 or 6 2 = | e 2 ; on the supposition that e l + e 2 + e 3 = ; and e-j = e 2 + 2H, e 2 = 2e 2 , e 3 = e 2 2^. s -T (s-ejda- fA s -T (s-e J JSJ^-^- ei.or-e^o e s ) cr where 2 = 4(<r e 1 )(cr e 2 )(o- e 3 ) = 4<7 3 y 2 o- y 3 , suppose ; and the discriminant A 7 of 2 is now positive. 60. Now, y 9 =-4(e 2 y 3 = 46^3 A 7 = y 2 3 - 27y 3 2 = 256^ 2 (4JJ 2 - 9<? 2 2 ) 2 . .,, , e. 7 -e 3 2#-3e 2 , 6l -e 9 Also with A 2 = * -3=a 2 , X 2 =^ -- ^ = e l~ 6 3 *" 6 1~ C 3 4ff ! _9^_^ 2 3 ~H 2 lA ~ Denoting by J the absolute invariant of 2, then ( 53) y 2 3 = 4 (1-X 2 X /2 ) 3 A 27 X 4 X /4 If we put 4X 2 \ /2 = 1/T, then (4 T / -1) 3 r 27T d while, with 4/cV 2 = r in (D), r_- 8 Now, if 2/c/c = 2XX , then T T /= 1, the relation which holds in the transformation from a negative discriminant in $ to a positive discriminant in 2. If we equate the values of J in (C) and (E), we find (1-fc) 2 & 1 G.E.F. 50 THE ELLIPTIC INTEGRALS. 61. When A is negative, and when we know the real factor s <? 2 of $; so that, with Je 2 2 + ri 2 = then, with # 2 = J(9e 2 2 + 4ii 2 ), and expressed as in 46, f ds I 8 -e 2 -H U=/ -7-3 = /rrCn- 1 yv, .............. (62) y x/>3 2*/-ti s e 2 + H with 2KK=n/H ; so that -- e 2 +H-(e 2 -H)cn(2u*/H) . , or p*= _J__ 2 ; (63) by means of which we change from Weierstrass s notation to Jacobi s and vice versa, when A is negative. Thus, for example, if # 2 = 0, then <? 2 = (J<7 3 )*, n 2 = f e 2 2 , # 2 = 3g 2 2 ; and, as in 46, 3 + l)ft.93) i Sinl5 } -lXi^)* / / ^g (4iM7s) = rl(8; -^ --^ sin 75-1- i S 62. Supposing s to range from oo to oo in the integral r> = /ds/^/S, when A is negative, then * (i.) <x> where o> 2 denotes /du/tJS, the real half period of n. (ii.) e 2 >s>-cc, where w/ denotes /ds/^S, a pure imaginary quantity, called the imaginary half period of $m ; and the period parallelogram ( 55) is now bounded by o> 2 and , , as adjacent sides. Also (47), mrKIJ-H^-iriJH. ,.-(66) THE ELLIPTIC INTEGRALS. 51 63. Treating in the same way the integral (2), _ /"" dx I /(~\ _ /r 2 *- 2 4- i^rV */ /^/V- 1 - * K ** ^ by replacing z by 1 z 2 in 38, 55 ; (i.) oo>a>l, (ii.) (68) (iii.) 1 > x> co , u= iK + 2K+icn-\-l/x, K) .......... (69) 64. By the substitution x 2 = l/y, the integral dx " dy i r-ds 7 ~^AJ 7S " on putting y = s \B/A ; which can be expressed by Weier- strass notation, or by the notation of Jacobi, when the factors of the denominator are known, as in equations (12) to (19) ; f_ E+Fx _, J J(A + Ba? + Cx* + Dx Q ) can thus be reduced to elliptic integrals, of the form considered in 39-61, the first term by the substitution x* = I/y, and the second term by the substitution x- = z. rnu a a*dr a rhus the integration required in the rectification of 7 13 = a 3 cos W. But by substituting ? >2 /a 2 = l/y, we find a s dr r ady s = o so that 52 THE ELLIPTIC INTEGRALS. 65. Write X for ce 2 - a 2 . x 2 - b 2 . x 2 - c 2 , where a 2 >6 2 >c 2 ; and write Jf for b^/(a 2 c 2 ) ; then we find, on substituting y for I/O) 2 , and taking a, /3, y for 1/c 2 , 1/6 2 , I/a 2 ; (i.) oo > x 2 > a 2 , comparing with equation (18), fMdx_ l la- 2 -x~ 2 _ Ib 2 .x 2 -a 2 J -jx~ VF*=~ar 2 ~ ^a 2 .x*-b 2 Ia 2 -b 2 .x 2 Ia 2 -b 2 .x 2 -c 2 = cn " ~ 1 - to modulus (ii.) a 2 > a; 2 > 6 2 , comparing with (17) and (16), Mdx x /6 2 .a 2 -^ 2 T" Va 2 -6 2 .o) 2 ~ to modulus (iii.) b 2 >x 2 > c 2 , on comparison with (15) and (14), b Mdx f J " ~ /-* J ** *.a?-x 2 7 - 2 .......... (75) to modulus (iv.) c 2 > a 2 > 0, on comparison with (13) and (12), f Mdx 1 lb*.c*-x z J J(-X)- Vc 2 .6 2 -a5 2 THE ELLIPTIC INTEGRALS. 53 f x Mdx 1 Ia z -c 2 .x* J /(-*)" Vc 2 .a 2 -a 2 /a 2 .c 2 -o; 2 Ia?.b 2 -x* = cn- 1 A /^ ^ H = dn- 1 /x /r; ? 5 , (77) \c 2 .a 2 a? 2 yb 2 .a? x 2 2 *- 2 to modulus 66. When X is a quartic function of x, and we know a factor, x a, of X, then the substitution x a = l/y reduces JclxJ^/X to the form M/dy/^/Y, where Fis a cubic function of y\ and this form can be treated by the preceding rules. But, independently, if we can resolve X into four real linear factors, x a, x /3. x y, x S, so that X = x a . x /3 . x y . x 3, and we suppose that a > /3 > y > S ; then with (i.) oo > x > a, dx . . ( 78) indicating that we must put . 9 /3 S.X a 9j sm 2 = " 5 - 5 , cos 2 = 9 S.X a 9j a /3.x 2 " - 2 - - to reduce the integral to the standard form ( 4) 2 and then a y .p o the anharmonic ratio of the four points A,B,C, D, the >cte of the integral ( 54), given by x = a, /8 y, 5. The verification by differentiation is a useful exercise for the student. 54 THE ELLIPTIC INTEGRALS. (ii.) With a > x > /3, we change the sign of X to make the integral real; and now, writing M for J^/(ct-y./3-<S) throughout, r a Mdx -i /^L^- i /q-<?-a-ft_ i i la-S.x-y ^a-$.x-S~ y a -/3.x-S~ Va-y.a-<T" r*Mda I 8n -^l^^^ but now the modulus K is the complementary modulus to /c, so j. i j /o 7 / ct ~~ o . *y ~~ o tnat K K = ^ ; a y . p ^ the different forms of the result indicate the appropriate substi tution required for reducing the integral to the Legendrian form. (iii.) With /3 > x > y, X is again positive, and / X l la-y./3-X . la-6.X-y , . la-6.X-S /01 . = sn- 1 A /-^- ! ^ = cn- 1 A /-> r J - J: =dn- 1 A/ )0 % ,....(81) \/3-y.a- ^l/3-y.a-x Vp-3.a-X /* Mdx ^ X l/3-S.x-y ly-S./3-X ly-S.a-X -sn-M^ -isBcn-Mi -^ = dn- 1 -^- - V p-y.X-o Vp-y.aJ-o, with the same modulus AC as in (78). (iv.) With y>x> S, X is negative, and Mdx -S.8-x \a-y.8-x v y-S./3x yyS./3x \o-y.p- x dx -i la-y.x-S_ _ 1 la-S.y-x_ 1 ^_ 1 ja-S./S-x with the modulus of (79) and (80). la-S . /3-x , . A / j- ,....(84) \y-o.a-a; THE ELLIPTIC INTEGRALS. 55 (v.) With S>x> x , X is positive, and f 7^ "~4^ :: -~ (85) with the original modulus of (78), (81), and (82). 67. Landens Transformation. When Legendre s arid Jacobi s standard integral (1) is treated as a particular case of these integrals (81) and (82), we write a = l/X, = l,y= -!,<$= -1/X, so that Jlf=J(l+X)/X; and now, with y for variable, /l+X.1-3/ _ x /l-X.l + y_ /1-X.l + Xy , . V~2J^X - V 2.1-X - Vl+X.l-X-" (8 where the modulus AT is now given by /r = 4X/(l + X) 2 , so that and we are thus introduced to Landens transformation, to be discussed hereafter. Changing, in 41, x into y 2 , and k into X-, we find dy with modulus X ; indicating, on comparison with (86), results such as which can be translated into the various forms of Landen s quadric transformation. 56 THE ELLIPTIC INTEGRALS. Denoting integrals (86) and (88) by u and v, then ojj. y<y, i\i _ ^, i-xy cn 2 (v, X)=-jJ^y4, dn 2 (v, \) ss ~^- t (91) whence sn(v, X) = ( 1 + g/ M^>^)cnK ^ etc (92 ^ We can easily prove, or verify by differentiation, that = _ - _ -- 2 ^ to the same modulus /c = 2 x /X/(l+X); so that, denoting this integral by u, and denoting sn(w, K) by x, then - y or dn(w> K ) = , ; nd(M;/c) = .-. f -- 1 A since ^/ sn (^ ^) where v = (1 +O M 5 and thence X)nd(^,jc) J ............... (96) ^) n d(^/c); ............... (97) (Cayley, Elliptic Functions, p. 183). The relation (92) between x and ^/, namely, thus leads to the differential relation dx (98) THE ELLIPTIC INTEGRALS. 57 68. The six anharmonic ratios of a, /5, y, S, arising by per mutation or substitution, give rise to six values of the modulus k, given by ^i-^ rV-iL^i ................... (99) or sin 2 #, cec 2 #, cos 2 #, sec 2 #, -cot 2 $, -tan 2 #, iffc = sin 2 0; or tanh%, coth%, sech 2 u, coslrw, - cech 2 u, - sinh 2 u, if k = tanh 2 u. We may notice that the expression for / in (D) of 53 is unaltered if for k we substitute any of these other five values ; and, on comparison with Weierstrass s notation, J so that we may put - nom ~ A = 25(5 -" (1 and then ^ = T V(2-Q, 6 2 = T V(-l + 2fc), e 3 = r V(-l-&) ; so that h = (e. 2 e^l(e l e^) ) as in 51. 69. Degenerate Forms of the Elliptic Integral. When two of the roots a, /3, y, S become equal, the corre sponding integrals degenerate into circular and hyperbolic integrals, which can easily be written down, on noticing as before ( 48) that (i.) when k = 0, STL~ I X becomes sin" 1 ^, cn^se becomes cos" 1 ^, etc; (ii.) when k=l, sn 1 ^ becomes tanh" 1 ^, cn" 1 ^ or dn -1 ^ becomes sech^, and tn~ 1 o? becomes sinh" 1 ^. When two of them are equal, we may replace the four, quantities a, ft y, S by the three distinct quantities a, 6, c, suppose, where a > b > c ; and now the degenerate elliptic integrals fall into three classes, I., II., III. I. Writing M for J^/(a b . a c) ; then (i.) oc > x > a, f Mdx . , , la b.x c , , la c.x b IT - r - 77 -- j - r = 81DJl- 1 A/T - ~ = COSh ~ 1 /v / j - . J (x a)tj(x b,x c) \b-c.x a \lb-c.x a (ii.) a>x>b, f x Mdx , la b.x c . , , la c.x b /- - - 77 - T - r = cosh~ 1 /v / I - = sinh~ 1 /v / I - . J (a x)fj(x b.x c) yb c.a x yb c.ax b (iii.) b > x > c, Mdx _ _ _ 1 la b.x c_ . _ 1 la c.b x (b-x.x-c)~ Vb-c.a-x- ~ ^Jb-c.a-x 58 THE ELLIPTIC INTEGRALS. c.b r x Mdx . 1 la b.x c // - \ /,-L \ = sin ~ \ rr~ = cos J (a x)*J(b x.x c) ^b c.a x c (iv.) c>x> oo , C c Mdx . , la b.c x la c.b x /7 -- \ 77i r = sinn~ l \hr- - = cosh~ l *- 1 . J (a x)^/(b x:c x) \b c.a x yb c.a x X II. Writing M for ^(a-b.b-c); then (i.) x > x > a, C x Mdx . , Ib c.x a la b.x c h - IA // v= sin \\ 7 = cos" 1 ,*/ 7. J (x b)^/(x a.x c) ya c.x b \a c.x b a (ii.) a>x>b, f a Mdx . Ib c.a x , . la b.x c If - 7\ // x = smh~ 1 A / T = cosn~ 1 A/ r. J \ x ~ b)*J(a x.x c) ^la c.x b \ja-c. x b X (iii.) b > x > c, f x Mdx , , Ib c.a x la b.x c /7I - \~~77 - r = cosn- 1 A/ - -. - = smh~ 1 A / - j - . J (b x)+/(a x.x c) \a c.b x \a c.b x c (iv.) c >x> oo, /" c Mdx , Ib c.a x la b.cx /7L - T~77 \ = COS" 1 ^/ - j - = ffln Vl - 1 - . J (o x)^/(a x.c x) ^a c.b x \a c.b x III. Writing . for %J(a-c. b-c) ; then (i.) oo > a; > a, . la c.x b . , , Ib c.x a - 1 - = - 1 -- r r . - r - A - j A -- r . c)^/(x a.x b) ya b.x c \a b.x c (ii.) a>x>b, a Mdx la c.x b . , Ib c.a x -- r~77 - f^ = cos- 1 ^/ - T - =sin~ 1 /v / - ^ - ; x c)^(a x.x b) \a b.x c y a b.x c x Mdx _ _ . _j la c.x b_ _ a Ib c.a x -c)J(a-x.x-b)~ <\/a-b.x-c~ ~ ^a-b.x-c b (iii.) b >x> c, f b Mdx . la c.b x Ib c.a x 77 - \ 77 - L - r = Sinh~ 1 A / -- r - = COsh~ 1 A / --- T - . J(x-c} t j(a-x.b-x} ^Ja-b.x-c ^a-b.x-c X (iv.) c > x > oo , r Mdx , . la c.b x . , , Ib c.ax i -- - - = cosh" 1 A / - j = smh" 1 A/ -- T -- . J (c x)*J(a x.b x) ya b.c x ya b.c x THE ELLIPTIC INTEGRALS. 70. When all four roots of the quartic X = are imaginary, so that (x - a)(x -/3) = (x- m) 2 + n z , (x - -y)(x -S) = (x- is reduced by the substitution Let us suppose that X is resolved into two quadratic factors, so that X is of the form X = (ax 2 + 2bx + c)(Ax 2 + 2Bx + C), where, by supposition, ac b 2 and ACB 2 are negative, so that the roots of X = Q are all imaginary. then the maximum and minimum of y, the tui-ning points of y, being denoted by y l and y.-,, X-L and x. 2 denoting the values of x cormsponding to y l and of y ; and now dy _2(Ab- aB)( Xl - x)(x - x 9 ) dx (Ax 2 + 2Bx + C) 2 For x is given in terms of y by the solution of (Ay-a)x- + 2(By-b)x + Cy-c = 0, ............ (104) and this equation has equal roots at the turning points of y, which are therefore given by the quadratic equation or (AC-&)tf*-(Ac+aC-2Bb)y+ac-&=O t ...... (105) and then B-b ax + b._ bx+c -a y y ~Ax+B~Bx+CT Ddy dy 2(Ab-aB) and (Ay 1 -a)(a-Ay. z )= - AC-B 2 60 THE ELLIPTIC INTEGRALS. sothat , . -, , ,,-x f which, by (lo), gives / * -i /=1L -}- cn-\ /^-4-dn-i / V 2/i -2/2 x/2/i v 2/i -2/2 N/2/i >2/i with * 2 = 1 - y 2 /y v K " 2 = the last expression, by the inverse dn function, being the simplest, as expressing a function of an argument oscillating between two positive limits, y^ and y 2 . 71. For example, if X = x 4 + 2a 2 o; 2 cos 2a + a 4 = (x 2 + 2ax sin a + a?)(x 2 - lax sin a + a 2 ), and if y (x 2 + 2a& sin a + a 2 )/(a? 2 2acc sin a + a 2 ), then #! = , 2/i = tan 2 (j7r + Ja) ; X 2 = a, 2/ 2 = tan 2 (j7r Ja); so that // = tan 2 (^7T Ja) = (1 sin a)/(l + sin a) ; r _ dx J V(^ + ^a^c^cos 2a + a 4 ) , a 2 (l+sina) But, by substituting ~2 = ^j -- > ct I r J .. . x , /-, A r\\ 5 -en- 1 (3; sma) = 7r-cn- 1 - T - -5, ...... (109) 2a 2a 2 2 by (2), a reduction of the elliptic integral to a different modulus, the modular angle being now a ; affording another illustration of Landen s transformation of 07. Thus, with a = JTT, equation (108) gives where K = (J2-I) 2 (when K \K=\} ; and by (109), dx ,l x 2 etc - For other numerical examples, the student may take ; 2 + 3, etc. THE ELLIPTIC INTEGRALS. 61 72. When two roots only of the quartic X = are imaginary, we may still make use of the substitution ( 70) y = N/D, where X = ND\ but now take ac b 2 negative, and ACB 2 positive. Proceeding as before we find that the maximum y l is positive, but the minimum y 3 is negative ; and y oscillates between and y l for real values of ^/X ; and /dx _ 1 7? ~ J(AC-B*) so that, by (14), i, ...(110) ~2/3 with K 2 = ^(^ - 2/3). * /2 = - 2/&%i - 2/s). 73. By another method of reduction we shall find (Enneper, Elliptische Functionen, p. 23) dx x - a) 1 ^ /j^ _ ^ J \/{a-x.x-/3.(x- _ lZ(a-x)-H(x-/3) , etc. ; where R 2 =( a - m) 2 + n 2 , K 2 =(/3- m) 2 + n z ; and Ac 2 = i-a-3 2 -^ 2 - so that ZKK =n(a- 13) /H K. Degenerate forms occur when a and /3 are equal ; and now dx n(x-a) dx THE ELLIPTIC INTEGRALS. 74. Replacing y by N/D in equations (102), then N- Dy. 2 = (a- Ay.^)(x - x 2 ) 2 ; so that we may write, according to Mr. R. Russell, D = Ax 2 + 2Bx + C = P(x^ - x) 2 + Q(x - x 2 ) 2 , N=ax 2 +2bx+c=p(x 1 -x) 2 +q(x-x 2 ) 2 ; .......... (113) where P = (Ay l -a)/(y 1 -y.J, Q = (a-Ay 2 )/(y l -y 2 ); and p=Py 2 , q = Qy r Interesting numerical examples can be constructed by giving arbitrary integral values to x v x. 2 , P, Q, p, q ; and now the substitution ^ = ^H^2 , fda _C J sJX J will make, as in 37, ^ J(p+q!?.P+Qz*r 75. When the factors of the quartic X are unknown, we employ Weierstrass s function, and we shall show subsequently in Chap. IV. that the elliptic integral Jdx/^/X is reduced to Weierstrass s canonical form \Jdsj ^S ( 50) by the substitution s=-H/X, H denoting the Hessian of the quartic X (Cayley, Elliptic Functions, p. 346) ; we may thus write ~dx T ,/ H = where g 2 , g 3 are the quadrinvariant and cubinvariant of the quartic X or aa) 4 + 46x 3 + 6c so that g 2 = ae ^bd + 3c 2 , g B = ace + 2bcd ad 2 eb 2 c 3 , H = (ac - 6 2 X + 2(ad - bc)x 3 +( and the general reduction of the elliptic integral of the first kind Jllxj^/X, where X is a cubic or quartic function of x, is now complete. The application of this general method to the particular cases already discussed is left as an exercise for the student. 76. Systematic Tables of the integrals of the elliptic functions sn u, en u, dn u t ns u, ds u, cs u, dc u, nc u, sc u, cd u, sd u, nd u, and of their powers have been given by Glaisher (Messenger of Mathematics, 1881 ). THE ELLIPTIC INTEGRALS. 63 Suppose yen udu is required ; we may write it o cnudnudu r cZsnu 1 . 1 =/ 771 - 9 ^-^ = -sm- dnu yV(l JK%n%) /c o etc. ; so that IK. en -M/cZu = cos " x (dn u) = sin ~ 1 (/c sn u) = tan ~ 1 (/c sn u/dn u) o = J sin " I (%K sn t& dn u) = am(/ci6, l//c), etc. Similarly, K u + KCiiu , dnu+/ccnu , K =log -i =logj , etc., K dnu /ccnu while /dn udu = cos " 1 (cn u) = sin ~ 1 (sn u) = am ^ .......... (1 1 6) o As an exercise the student may integrate nsu, dsu, ...; also sn 3 i6, cn%, dn 3 u, ...; and obtain formulas of reduction for the integrals of (sn u) n , (en u) n , (dn u) n , As a general method, for (snu} n for instance, we put sn 2 u = s ; and now s.i-k S r Un suppose - By means of the well known formula of reduction, for v p =Jx p dxj^/N, where N=ax* + 2 we have, on comparison, a = fc, 6= -1(1 + ^,6=1,^ = 1(71-1); so that v p = 2u n , v p+l = 2u n+ to v p - 1 = 2u n - 2 , and (n + l)fcie n+2 ?i(l + ^)u n + (?i l)u n - o = sn n ~ l u en u dnu,...(l 1 7) the formula of reduction for u n =y(sn u) n du. When the limits are and K, we obtain the recurring formula (?i+i)^ /l+2 -?i(i+^K+(^-iK- 2 =o, ...... (118) ri-n- analogous to Wallis s formulas for /(sin or cos 0) n dO. o The same formulas hold for u n = (cd u) n du, since ( 57) cdu = sn(^ u). Thus u n is made to depend ultimately on u v already deter mined, or on u. 2 ; and a similar procedure will hold for the integrals of (en u) n or (sd u) n , (dn -u-) n or (nd u) n , etc. (54 THE ELLIPTIC INTEGRALS. 77. The Elliptic Integral of the Second Kind. We may mention here incidentally that the integrals of sn 2 u, cn 2 u, dn 2 u, us 2 u, ds 2 u, cs 2 i&, . . . require for their expression new functions called elliptic in tegrals of the second kind, such as occur for instance in the rectification of the ellipse. For if, in the ellipse (cc/a) 2 +(2//6) 2 = l, we put x = a sin 0, y = b cos < ; then - = + = a 2 cos 2 </> + &2sinV = d\l - e 2 sin 2 0) ; Cup ct(p cicp" so that -=/J(l-e 2 sin 2 <j>)d<p =/A(</>, e)^-/dn 2 itcZ a o oo on putting = am(u, e); and 6, the excentricity of the ellipse, is now the modulus. The integral y^/(l /c -2 sin 2 0)(i0 oryX(0, K)d<f> is denoted by o JE((p, K) by Legendre, and called the elliptic integral of the second kind ; and when the upper limit is JTT, the integral is denoted by E I K, or by E simply, and called the complete elliptic integral of the second kind. Examples. The following examples are collected chiefly from Legendre s Functions Elliptiques ; the results, being now expressed by the inverse elliptic functions, will serve as a guide to the substitutions required to reduce the integrals to the standard elliptic forms, and the correctness can be tested by differentiation as an exercise. 4. /(x-a.x-/3)- idx THE ELLIPTIC FUNCTIONS. 6. 7. Prove that, if w n = 4>x n (I - x n ), 2- 1 / and express the result when n = 3, 4, or 6. 8. Prove that, if x a is a factor of the cubic X, so that X = (x - a )(ax 2 + 2bx + c) ; fv 7. 3 7^ ~aa 2 a an integral occurring in the determination of the motion of a projectile in a resisting medium. Evaluate the integral when acr + 2ba + c = 0, so that . Prove that (i.) A cn ^ = /1 k y l + dnu A/l -dnu x.. . r K snudu _ 1 J dnu + K K (!+K ) o /*8JC (iii.) /u sn%<iu = 2K(K-E)/ K 2 . o /-Jir 1 (iv.) /f\<A, /c)sin 6<i^> = - sin -^ J K 10. Prove that ^//c /2 >K>E> 11. Denoting the integral f(^)- n d^ by u n , establish the formula of reduction /c % w+2 - (n - 1) (1 + K* 2 )u n + (n- 2)u n _ 2 = - /rsin cos Evaluate -u w for -w. = 2, 3, 4, . . . . G.E.F. CHAPTER III. GEOMETRICAL AND MECHANICAL ILLUSTRATIONS OF THE ELLIPTIC FUNCTIONS. 78. Graphs of the Elliptic Functions. Now that the Elliptic Functions have been defined and a few of their fundamental properties have been established in Chapter I. in connexion with the pendulum; while in Chap ter II. the reductions of the elliptic integral to the standard form have been tabulated, let us consider some further applica tions, and first in connexion with the graphs of am u, en u, sn it, dn u, represented by curves whose equations are of the form y am x, en x, sn x, or dn x. The graphs of these equations are given in fig. 5, in curves (i.), (ii.), (iii.), (iv.) ; the modular angle employed is 45, so that the curves can be plotted from the numerical values given in Table II., analogous to the graphs of the circular and hyper bolic functions, given in Chrystal s Algebra, Part II. ; thus, for instance, the curve y = &mx is the graph of the relation between $ and u in 5. We notice from the equations of 57, Chap. *IL, that by sliding the curves along Ox through a distance K, the curve y = snx becomes changed into y = sn(I -f #) = cn#/dn# or cdx, and not into y = cnx ; while the curve y = cnx becomes changed into y = cn(x K) = KS n.x/dux or tc sdx, and not into y = snx; so that the curves y = sn x and y = cnx are essentially distinct curves, and cannot be superposed, like y = cosx and y = sinx. The curve (i.), the graph of am x } consists of a regular un dulation, running along the straight line y = \irx\K\ so that am x = ^7rX/K-}- periodic terms = \-wx\K- THE ELLIPTIC FUNCTIONS. 67 in a Fourier series, where the B s are to be determined sub sequently ; and then by differentiation, dn x = (k-jr/K) { 1 + 22m,B n cos(nirxlK) }. So also the graph of E$ or Eamu, the elliptic integral of the second kind ( 77) consists, like (i.) the graph of aino:, of an undulation running along the straight line y = Ex/K; so that we may write, in Jacobi s notation, where Zx is a periodic function of x, which can be expressed in a Fourier series and then, by differentiation, dn 2 # = EjK+(irlK)I,nC n cos n-n-x/K ; whence also the expression for sn 2 # and cu-x in a Fourier series. Fig. 5. We proceed now to some mechanical and geometrical appli cations of these curves. 79. PROBLEM I. The curve assumed by a revolving chain We shall prove that y/b = sn Kxlci (fig. 5, iii.) is the equation of the curve of a uniform chain, rotating steadily with constant angular velocity n about an axis Ox, to which the chain is fixed at two points, 2ct feet apart, gravity being left out of account, e.g. a skipping rope. 68 ILLUSTRATIONS OF Denote by t the tension in poundals of the chain at any point, and by w the weight in Ib. per foot of the chain. Then the equations to be satisfied are d / dx\ _ . d / dy ds\ ds/~ ds\ ds. Therefore tdx/ds = T,a, constant, the thrust in poundals in the axis due to the pull of the chain ; and therefore rJ /rlii\ /n^-fin fl^ti rJw (.(/ / (A/U \ Iv W tt/t/ U/JU the differential equation of the curve of the chain. But 14-^ 2 l so that dydty dsd?8 dx dx z dx and therefore Integrating, supposing y = b when dy/dx = and ds/dx ds n 2 iv., 9 9 . ^=l + w (^-^); so that t = Tds/dx = T+ i% 2 w(6 2 - y-). so that x is an elliptic integral of y, of the form (5) in Chap. II. ; and y is an elliptic function of x, obtained by inverting the function of the integral. To obtain this function, let y = b sin < ; then ri 2 wb- ,1 T rX i_ /c nw so that d> = am K-, where = ^ ; a a 2T and y/b = sn Kx/a, the equation of the curve formed by the chain ; and now 2a denotes the distance between the ends of the chain. THE ELLIPTIC FUNCTIONS. 59 We may denote Tj\n 2 w by Ji 2 ; and now h- K " 2 _ti 2 lab , la . , 19. ~?~ro J^K whence the modulus K and quarter period K can be determined when h and a are given ; and while ^ = ~A0; fix a and integrating, with the notation of 5 and 77, Kh* If 2^ denotes the length of the chain, then s = l when = 1^, and J^(0, *)=. , j^(0, /c)=^; and therefore I + a = JJ?JST^/ = ^ = 2a Ay^ /2 , from which /c, -ST, and .fi 1 must be found by a tentative process, from Legendre s F.E., II., Table II., when a and I are given. For instance, if K = K = |^/- as ^ n ^ a ble II., page 11, A"= 1-85407, ^ and 6/a = 1 5255, l/a = l 92 80. When the chain is fixed at two points not in the axis, nor in the same plane through the axis, the chain when re volving in relative equilibrium will form a tortuous curve, which will sweep out a surface of revolution, of which the preceding curve y/b = snKx/a is a particular case of the meridian curve, while the general equation is of the form For in this more general case the equations of relative equilibrium are now d (,dx\ d ( dy\ , d (,dz\ -y-U-f ) = 0, -^-i t-Y- } + n- W y = 0, -y-f t-j- + n*wz = 0. ds\ ds/ ds\ dsJ ds\ dsJ Three first integrals of these equations are H, a constant; (2) and t-\-^n-iv(y 2 + z 2 ) = \, a constant (3) 70 ILLUSTRATIONS OF Putting y* + z z = r z , th dy dz _ .dr 2 y dx ^dx~^dx and from (1) and (2), 2/^-^| = f ; therefore, squaring and adding, d < d& or = 2 (|! - 4) - -*^* = ^(2X - suppose ; arid for r 2 to lie between b 2 and c 2 , we must suppose <i 2 > fr 2 > r 2 > c 2 , and as it is of the form (17), p. 37, we put W - r 2 = (6 2 - c 2 )cos 2 0, r 2 - c 2 = (6 2 - c 2 )sin 2 0, cZ 2 - r 2 = d z - c 2 - (6 2 - c 2 )sin 2 = (d z - c 2 ) A 2 0, where /c 2 = (6 2 -c 2 )/(rZ 2 -c 2 ). Then - c 2 )cos 2 sin 2 or . r = so that <p = am Kx/a, where K z /a z = n*w 2 (d 2 - c 2 )/4T 2 = 4(cZ 2 - c 2 )/h 4 ; and then r 2 = y 2 + z 2 = b^r^Kxja + c 2 cr\ 2 Kx/a, the equation of the surface swept out by the chain, the meridian curve being similar to curve (iv.) in fig. 5. 81. The chain will obviously take up the form which, with given length between the two fixed ends, has the maximum moment of inertia about the axis of revolution ; and we have thus investigated the solution of an interesting problem in the Calculus of Variations. THE ELLIPTIC FUNCTIONS. 71 The form of the chain for a minimum moment of inertia is obtained by supposing that r 2 >d 2 , as in (13), p. 35 ; and by putting r 2 d 2 = (d 2 6 2 )tan 2 ^>, T 2 -b 2 = (d 2 -b 2 )sec 2 <j>, r 2 - c 2 = (d 2 - c 2 ) A 2 sec 2 0, K 2 = (b 2 - c 2 )/(d 2 -c 2 ), as before. Then ~ 2 = 4(d 2 - 6 2 ) 2 tan 2 - b 2 ) 2 (d 2 - c 2 )tan 2 sec 4 A 2 0, so that <p = am Kx/a, and then y 2 + z 2 = d 2 sQC 2 <j> - 6 2 tan 2 a - b 2 sc 2 Kx/a is the equation of the surface of revolution upon which the chain lies, when its moment of inertia about the axis of x is a minimum. The projection of the chain upon a plane perpendicular to the axis is to be investigated subsequently. 82. When the two points to which the ends of the chain are fastened lie in the axis, or in a plane through the axis, the chain takes the form of a plane curve, whose equation is y/b = sn Kx/a for a maximum moment of inertia, as already shown in 79 ; and y en Kxja = cZ, or y = d nc Kx/a for a minimum moment of inertia ; which can be proved as a simple exercise in the Calculus of Variations, by considering the variation of the integral 83. PROBLEM II. " The curve on which an ellipse, of semi- axes a and b, must roll for its centre to describe a straight line Ox is the curve whose equation is y/a = dn x/b, the modulus /c being the excentricity of the ellipse." 72 ILLUSTRATIONS OF For if the centre M of the ellipse describes the horizontal straight line Ox (fig. 6), M must always lie vertically over P, the point of contact with the fixed curve, so that the ellipse rests in neutral equilibrium if its centre of gravity is at the centre M\ teeth being cut in the curves, if requisite, to prevent slipping. Therefore the polar subnormal ^ n dr . ,. ... 1 cos 2 # sin 2 $ MG= " m the elhpse ~^~^~~ dO must be equal to the subnormal = y^r i* 1 the fixed curve AP } where MP = r = y. Fig. 6. Now in the ellipse, differentiating, 2 dr (I 1\ . . I/I 1 1 1 -- * ^Z == lf2 -- o)2sm0cos0=2 A /( 9 -- 5-r i r 3 dO \6 2 a 2 / \ \r 2 a 2 6 2 r 2 snce or - 9 r 2 a 2 ^ o 2 7^ 2 6 2 a 2 / so that in the fixed curve AP . dy by (9), p. 33 ; or, by inversion of the function, y/a = dn x/b. THE ELLIPTIC FUNCTIONS. 73 The arc of the rolling carve is obviously the same function of r as the arc of the fixed curve is of y ; and therefore the arcs are expressible by elliptic integrals of the second kind. The curve AP can be described as a roulette, by a point P fixed to a certain curve which rolls on Ox, and therefore touches Ox at G, since G, the foot of the normal PG, is the centre of instantaneous rotation. Since PM is the perpendicular from a pole P on the tangent of the rolling curve, and that the relative orbit of P and M is the ellipse, therefore the pedal of the rolling curve with respect to the pole P is an ellipse ; or, in other words, the rolling curve is the first negative pedal of an ellipse with respect to its centre, that is, the envelope of lines drawn through each point on the ellipse perpendicular to the line joining the point to the centre of the ellipse. The first negative pedal of an ellipse with respect to its centre is called Talbot s curve ; its (p, CD) equation is 1 _ cos 2 co sin 2 o> p~~~oT ~P and it is of the sixth degree (Cayley, Proc. R. S., 1857-9, p. 171). 84, For a rolling hyperbola, changing the sign of b 2 , the fixed curve must be given by abdy ab r_ -J J (y *- by (8), p. 33 ; so that, by inversion of the function. a/y = en X/CLK, or y/a = nc X/CLK, is the equation of the fixed curve for the hyperbola. 85. When the fixed curves are of the form of curves (ii.) and (iii.) in fig. 5, we shall find in a similar manner that the rolling- curves which will rest upon them in neutral equilibrium are given by 1 cosh 2 , sinh 2 # 1 cosh 2 sinh 2 _ - ; _ j _ /"1 1* __ - - _ _^ _ r 2 a 2 b 2 r 2 a 2 b 2 Taking the first of these two rolling curves, 74 ILLUSTRATIONS OF _ dr _ ? V(a 2 - r 2 . b 2 + r 8 ) d0~ ~^T~ so that in the corresponding fixed curve dx ab x -f a M y ab fy Jj(a*-y2.b 2 + y 2 ) T^hP) U by (7), p. 33 ; so that, by inversion, y/a = en #/&/c, with mod. K = a/^/(a 2 + 6 2 ). Similarly it can be proved that the second rolling curve can rest in neutral equilibrium on the fixed curve (fig. 5, iii.) y/a = sn xja, with mod. a/b. 86. TROBLEM III. Dynamical Problem. "The curve rcuO = c is the relative orbit of the centres of gravity of a straight rod fitting into a smooth straight tube, resting on a smooth horizontal table, when struck by an impulsive couple, the centres of gravity of the rod and of the tube being initially c feet apart." Suppose the rod to weigh m Ib. and the tube to weigh M Ib., and denote the moments of inertia about the centres of gravity by mk 2 , MK 2 (Ib. ft. 2 ). Then, if P is the C.G. of the rod, Q of the tube (PQ = r), and the (stationary) C.G. of the system, Denoting by n the initial angular velocity communicated to the system by the impulsive couple, then from the Principle of the Conservation of Angular Momentum, {m(k 2 + OP 2 ) + M(K 2 + OQ*)}(de/dt), ( 79 MY mMr 2 \dO ( 72 , ,, T , 9 , mMc 2 \ /1X or 1 ^+XK*+--= m ] l *+MK*+n ..... (l) Again, from the Principle of the Conservation of Energy, m or, after reduction, 1 2 THE ELLIPTIC FUNCTIONS. 75 the kinetic energy in foot-poundals, is constant, and Therefore, employing the value of dOjdt given by (1), or, finally, + J/ZT 2 + mJ/c 2 / (m + Jl/) f so that r is an elliptic function of 0, given by (8), p. 33. We therefore put r = csec<; and then find where /c 2 = ^ T777- so that = am 0, cos < = en ; and therefore r en = c. 87. When c = 0, /c = l, and this method fails; but now r 2 dfl 2 = (ra& 2 + MK *)(M+ m) = a 2 suppose, where a 2 = (991 + M)(mk z + MK*)/mM ; /^ ft K* st and now = 7 77 rr-^ = sinh~ 1 -, J T^/( I -f r 2 /a 2 ) r or rsinh$ = , the equation of one of Cotes s spirals, the relative orbit of the centres of gravity of the rod and tube, ultimately described after leaving the unstable position of coincidence. The system of the rod and tube may be supposed started by any arbitrary impulse, not necessarily a couple, and the essential character of the relative motion is unaltered; but now the C.G. of the system is no longer at rest. 88. Other mechanical arrangements, leading to the same equations of motion, will readily suggest themselves ; thus the tube may be supposed to be one of the hollow spokes of a wheel of weight M lb., moveable about a fixed vertical axis, while the rod is one of a number of equal rods, or balls, of collective weight m lb., one in each tube, and initially placed with the C.G. at a distance c from the axis of the wheel. 76 ILLUSTRATIONS OF Now, if the wheel is started by an impulsive couple with angular velocity n, the path of the C.G. of each rod or ball in its spoke will be of the form r en (9 = c. 89. PROBLEM IV. Central Orbits and Catenaries expressed by Elliptic Functions. When a Central Orbit, expressed in the polar coordinates (1/u, 0), is described under an attraction to the pole, of magni tude P (dynes per gramme), then, as is proved in treatises on Dynamics, P is given by the equation 73 z.2 *fd Zu i \ i 7 40 1 dO P = h2 u 2l + u \ W h ere / 6==r 2 = \d0 2 J dt u 2 at and the constant h is twice the rate of area swept out by the radius vector ; and v the velocity is given by h 2 ,jd Given the equation of the orbit as a relation between u and 0, the value of P as a function of u is thence easily determined by differentiation, as in 30 ; let us then determine P for the orbits au = sn, en, tn, or dn m6 ; also for the inverse curves au = ds, nc, cs, or nd m9, in Glaisher s notation ; the remaining orbits au = cd, sd, dc, ds mQ ; are not distinct curves, being merely formed by reflexion in the line 0=$K/m, since cd mO = 8i\(K mO) ( 57), etc. As in 30, we shall find by differentiation that (d 2 u/d6 2 ) + u is always of the form Au + Bu s , so that P is of the form /x,u 3 + vu 5 ; and conversely, given this form of P, we find by integration that (du/dO) 2 is of the form C+Du 2 + Eu* , so that is an elliptic integral of u, and u an elliptic function of (9, of which the results are given in 36. When the orbit is given by we find by differentiation, as in 30, that P is of the form Xu 2 + /xu 3 + j/u 4 ; and conversely, when P is of this form, (du/dO) 2 is a cubic form in u ; and 6 is given as an elliptic integral or inverse elliptic function of u, by the results of equations (12) to (45), Chap. II. THE ELLIPTIC FUNCTIONS. 77 As an exercise the student may determine the value of P and v\ as functions of u or r, in the orbit 1 _cn 2 m$ sn 2 ra0 and its inverse curve, whose equation is of the form Similarly the central forces required to make a chain assume the form of one of the preceding curves can also be determined (Biermann, Problemata quaedam mechanica functionum ellipticarum ope soluta, Berolini, 1865). When a transverse force T is introduced into the field of force, then h is no longer constant, but, as demonstrated in treatises on Dynamics and the Lunar Theory, - dh?_2T T _dlogA. ,., dP+ -~ *= V~ dO T du If we assume P = h 2 u 3 ; then d (, du\ dlosh ,du l s+ ~ => or h = c > a constant But = /m 2 , so that = Cn 2 } or =-C, which shows that the body approaches the centre with constant velocity C. Suppose, for instance, we take an orbit given by m6 = am au, then h = C = C ~ and P = hW = T= u*~ - = - CAin mO cos mO ; so that V, the potential of the field of force, is given by and then P= _, T= -. - 78 ILLUSTRATIONS OF 90. PEOBLEM V. The motion of Watt s Governor. "The oscillations of Watt s Governor between the inclina tions a and /3 to the vertical, when constrained to revolve with constant angular velocity co, are given by tan|$ = tanjadn(?i, K), with K = tan J/3/tan Ja, where 6 denotes the inclination of an arm to the vertical axis at the time t." Consider the motion of either rod and ball, as if unconstrained by the other, and denote by C the moment of inertia of the rod and ball about its axis of figure, and by A the moment of inertia about the axis on which the rod turns at the upper joint (fig. 7). Fig. 7. Drawing the three principal axes OA, OB, OC at 0, and three moving coordinate axes Ox, Oy, Oz, such that Ox and OA are coincident, Oz is vertical, and yOz, BOC in the same vertical plane, then the components of angular velocity about OA, OB, OC are (dO/dt), o>sin$, e*>cos$; and the corresponding components of angular momentum are -A(d6/dt), -4o>sin0, Coo cos 0. The components of angular momentum about Ox, Oy, Oz will therefore be fc 1= -A(d6/dt) t h 2 = (C-A)>sin6cos6, h s = (Ccos*6+Asin*6)a>; while the component angular velocities of the coordinate axes Ox, Oy, Oz are ^ = 0, 6. 2 = 0, 6 3 = u>, with the notation of Routh s Rigid Dynamics. Take the poundal as the unit of force, and denote by M the weight in Ib. of either arm and ball, by h the distance in feet from of the centre of gravity; the equation of motion THE ELLIPTIC FUNCTIONS. 79 obtained by taking moments about Ox or OA is or -A(d 2 0/dP) + (A-C)u 2 sin0cos0 = Mghsin0 , (1) so that, if A = C, the motion reduces to simple pendulum motion. Integrating, on the supposition that a > > j8, and that dO/dt = Q when = a and /3, rZ6 2 A G -jp = -j or(cos cos a)(cos /3 cos 0) (2) The position of relative equilibrium is given by d 2 0/dt 2 = Q; and then, if 9 = y, so that in these oscillations the point D, which controls the valve, makes equal excursions above and below its position of relative equilibrium. The technical name for these oscillations is "Hunting"; and some kind of frictional constraint is required to prevent these oscillations from becoming established. (Maxwell, Proc. R. S. t 1868.) Denoting tanja, tan J/3, tanj$ by a, b, x respectively, then equation (2) may be written 4 dx 2 _A-G pfl-x* l-a 2 \/l-6 2 1-; ~dP~ ~A~ w \T+^~T+~ or = cos a cos and this, by equation (9), p. 33, gives x = adn(nt, K), or tanj$ = where /c = 6/a = tan J/3/tan Ja, and ?? =o)sinjacosj/3 x /(l G For a small oscillation, we put a = /3; and then /c =l, /c = and now the period of an oscillation 27T 47T /_J L \ A ^ to sn a 91. If we suppose the whole weight of a rod and ball con centrated at the centre of gravity, we have (7=0, A=Mh-\ and now the motion may be assimilated to that of a particle in a smooth circular tube, which is made to rotate about a vertical diameter with constant angular velocity w. (Prof. B. Price, Analytical Mechanics, 403). 80 ILLUSTRATIONS OF The equation of motion (1) now reduces to h .. ., AcoVm cos = g sin 0, where h denotes the radius of the circle ; and for oscillations on one side of the vertical between a and /3, a > 9 > ft, (dO/dt) 2 = o) 2 (cos - cos a) (cos ft - cos 0), the solution of which is, as before, tan J0 = tan Ja dn nt, where /c = tan J/3/tanJa, n = w sin | a cos J/3. If the particle in its oscillations just reaches the lowest point of the circle, ft = ; and then K = 0, /c = 1 ; and now dnnt degenerates into sechnt ( 16) ; so that tan J$ = tan \a sech nt, where n = o> sin Ja ; the position of relative equilibrium being given by cos y = g/w 2 h = |(1 4- cos a) = cos 2 Ja. If the particle passes through the lowest point, it will come to rest again where 0= a; and now (dO/dt) 2 <o 2 (cos cos a)(2 cos y cos a cos 0), where 2 cos y cos a > 1; and the solution of this equation is tan J0 = tan Ja en nt, where n = ^^/(cos y cos a). When a = 7r, we shall find the motion given by so that, after an infinite time, the particle just reaches the highest point of the circle, where it will be in unstable equilibrium. A still greater velocity of the particle relative to the tube will make the particle perform complete revolutions, which will be expressed by ta,uW = Ctnnt. We have supposed the circular tube to be made to rotate with constant angular velocity about a vertical diameter ; but the motion of the particle relatively to the tube will be found to depend on similar equations when the tube is attached in any other manner to the vertical axis. THE ELLIPTIC FUNCTIONS. 81 92. Such will be the motion of a pendulum swinging about an axis fixed to the Earth, and now it is interesting to notice other cases of motion of bodies which can be directly compared and made to synchronize with the motion of an ordinary pendulum, swinging through a finite angle. Thus the pendulum, if moveable about a smooth vertical axis, which is fixed to a wheel moveable about a fixed vertical axis, the inertia of the wheel being sufficiently great for the reaction of the pendulum to have no sensible effect on its angular velocity, will perform pendulum oscillations, with g replaced by aw 2 , o> being the angular velocity of the wheel and a the distance between the axis of the wheel and of the pendulum. Again a cylinder of radius a and radius of gyration k, rolling inside a fixed horizontal cylinder of radius b, will synchronize with a pendulum of length l = (b a)(l+& 2 /a 2 ). If the fixed horizontal cylinder is free to rotate about its axis, and has its centre of gravity in the axis, then the length of the equivalent pendulum is -, , l=(b-a)(I+n\ where n = - 2 l+- 2 mk-, MK 2 denoting the moments of inertia about the axes of the rolling and fixed cylinders. The rolling cylinder may be replaced by a waggon on wheels, and the motion can still be compared with that of a pendulum. A circular cone, whose C.G. is in its axis of figure, and whose axis is a principal axis, performs pendulum oscillations when it rolls on an inclined plane, or inside or outside another fixed cone, whose axis is sloping, the vertices of the cones being coincident; the determination of I, the length of the equivalent pendulum, in these cases is left as an exercise to the student. In those cases where the finite oscillations are not of the pendulum character, we suppose the motion indefinitely small ; and now, in small oscillations under gravity, instead of giving the formula for the period of a small oscillation, it is in general simpler to give I, the length of the pendulum, whose small oscillations have the same period. G.E.F. 82 ILLUSTRATIONS OF Thus for the vertical oscillation of a carriage on springs, I is equal to the permanent average vertical deflection of the springs, due to the weight of the body of the carriage. For the small vertical oscillations of a ship, l=V/A, where V denotes the displacement of the ship (in cubic feet), and A the water line area (in square feet) ; and if the ship is floating in a dock of area B sq. feet, then it is easily proved that 93. The Reaction of the Axis of Suspension of a Pendulum. It is important to know the magnitude of this reaction in the case of a large swinging body, like a bell in a church tower. Denote by X and Y the horizontal and vertical components of this reaction, considered as acting on the swinging body ; and take the gravitation unit of force, the force of a pound. Then X, Fand W t applied at the centre of gravity G (fig. 1), will be the dynamical equivalents of the motion of the body, collected as a particle at G ; and since the component accelera tions of G are h(dO/dt) z in the direction GO, and h(d 2 0/dt 2 ) perpendicular to GO, therefore, resolving horizontally and vertically, Wh(d 2 0/dt 2 )cos 9- Wh(dO/dt) 2 sin = Xg, Wh(d 2 0/dt 2 )siu 0+ Wh(dO/dt) z cos Q=Yg-Wg- while, from the pendulum motion, I(d 2 0/dt*) =-g sin 0, l\d0/dt) 2 =g(2R-lvQrs 6). From these equations we find Y ^ 4>Rh 2h pp = 1 jsa&O + ,2- cos -/-cos 0(1 cos Q) t Y , h (2h r ~W ~~ + I = ~ VI x i-2h 4Rh\ .. an . TJ/- = I ~i --- /2/ sln r sm cos 0, and therefore the resultant of X and F TF(1 h/l) is a force in the direction GO ; and T varies as the depth of P below the line 2/ = i^ + i^> whence X and F are easily constructed. THE ELLIPTIC FUNCTIONS. 94. In the simple pendulum, h = I, and the tension T of the thread PO is given by At the end of a swing y = 2R, and T/W=l-2R/l; so that, if 2R is less than I, T is always positive. But if 2R is greater than I, so that the plummet swings through more than 180, T changes sign, and the thread will become slack, unless replaced by a light stiff rod. When 2R is greater than 21, the pendulum makes complete revolutions ; and now, at the top of a revolution, y = 21, and T/W=4<R/l 5 ; and when 2R is greater than -JZ, T is again always positive, and the plummet can be whirled round at the end of a thread, without the thread becoming slack. 95. When the axis of suspension of the pendulum is hori zontal, and cut into a smooth screw of pitch p, the equation of energy gives W(V + k*+p*)(dOldt) 2 = Wg(H-h vers 0), if the centre of gravity descends from a height H above its lowest position ; so that ( It,- + tf+p 2 )(d*Oldt 2 )=-ghsm 0, and therefore I = h + (k 2 + p 2 )/h ; and now in addition to X and F, the reaction of the axis exerts a horizontal longitudinal component Z and a couple pZ, given by W d 2 6_.-Wphsin6 ~ g P dt 2 ~ 2 Similarly the increase in I due to the pendulum being sup ported on friction wheels may be investigated. As an exercise the student may investigate the small oscil lations of a system of clockwork, in which the wheels are unbalanced about the axes, and prove that for small oscilla tions the length of the simple equivalent pendulum is given by I = (2wfcy)/(Zti&p*eoB a), where w denotes the weight, wh the moment, and iuk 2 the moment of inertia of a wheel about its axis ; a denoting the angle which the plane through the axis and centre of gravity makes with the vertical in the position of equilibrium ; and p denoting the velocity ratio of the wheel. 84 ILLUSTRATIONS OF 96. The Internal Stresses of a Swinging Body. These internal stresses are most forcibly realized on board a ship rolling in the sea, not only in their effects as producing sea-sickness, but also in causing the cargo to shift, if the cargo is grain, coal, or petroleum, in bulk. It is usual to consider the ship as acted upon by two forces, (i.) W tons, the weight or displacement of the ship, acting vertically downwards through the centre of gravity 6r, (ii.) W tons, the buoyancy of the water, acting vertically upwards through M the metacentre (fig. 8). t Fig. 8. These two forces form a couple of moment W.QM.smO (foot tons), so that the ship will roll about a horizontal longi tudinal axis through G, like a pendulum of length GL = k-jGM feet, Wk 2 denoting the moment of inertia of the ship about this axis of rotation. Now to find the force which acts upon w, any infinitesimal part at P of the ship, to give it its acceleration and to balance its weight, we refer the point P to axes Gx and Gy, drawn upwards through GM and perpendicular to GM. THE ELLIPTIC FUNCTIONS. 85 This force will balance the reversed effective force of w at P and the effect of gravity on w ; and therefore, in gravitation measure, will have components w d*9 w fd&\* , ~~ V "7/2 x \J+ ) ~^~ w cos ft parallel to Gx, w d 2 6 w fdO\ 2 ~"g X ~dP~ g V (di) +w sm P arallel to G V- If w is suspended as a plummet by a very short thread, the thread will take the direction of this force, and will therefore make an angle with Gx _ i9 s i n Q _ x(d z O/dt*) - y(dO/dtf gcos6 + y(d*0!dP) - x(d6/dty 2 Supposing the ship to roll like a pendulum of length , through an angle 2a, then I(d 2 0/dt 2 )=- -g sin 0, and ^I(d0ldt) 2 =g(cos6-cos a) ; and by 8, dW/dP = - n 2 sm 6=- 2- /i 2 sin J0 cos J0 = - *n* K sn nt dn nt, (dO/df = 2n\cos - cos a) = 4?i 2 (sin 2 Ja - sin 2 J#) = 47iVcn%l At any instant the lines of reversed resultant acceleration will be equiangular spirals, of radial angle 0, round the centre of acceleration G as pole, the resultant acceleration at P being /* . r r g,- sin cosec 0, and the resultant effective force w-j sin 6 cosec 0, when we put GP = r, and I(d6/dty=g sinOcot 0; so that tan = (sn nt dn nt)/(2ic cn 2 n). Superposing the effect of gravity, the resultant lines of force or internal stress will be equiangular spirals of the same radial angle 0, round a pole J, the position of which is obtained as follows (fig. 8) : Draw LK perpendicular to GL to meet the horizontal line GK in K; describe the circle on GK as diameter, and draw KJ making an angle GKJ=<}> with GK; this will meet the circle in /. For the resultant effective force of w at P, being /= iv-rsm 6 cosec = WTTJ> making an angle with GP t will, when compounded with w upwards, and taking the triangle PGJ turned through an angle as the triangle of forces, have a resultant t = w.PJjGJ, making an angle with JP. 86 ILLUSTRATIONS OF This will be the tension and in the direction of a short thread, from which w is suspended as a plummet at any point P ; and the deflection of this plumb line from its original mean direc tion in the ship will be a measure of the tendency of a body to slide or of a grain cargo to shift ; and to a certain extent of the tendency to sea-sickness at this point of the ship and at this instant of its motion. The tendency will clearly have its maximum value at the end of a roll, when dO/dt = 0, and = JTT, and then / coincides with K. (Prof. P. Jenkins, On the Shifting of Cargoes, Trans actions of the Institute of Naval Architects, 1887.) The plumb line at P will now set itself at right angles to KP, while the surface of water in a tumbler at P will pass through K ; and a granular substance at P will begin to slip if KP makes with its surface an angle greater than the angle of repose of this grain. Thus up the mast, at a distance a feet from G, water would be spilt out of a tumbler, or sand in a box would shift, by the rolling of the ship through an angle 2, which would not spill or shift, if the ship heeled over steadily, until an inclination /3 (the angle of repose of the sand) was reached, given by tan /3 = (1 + a/7)tan . At the centre of oscillation L, where a=l, there is no tendency for the water to spill, and this shows that the motion of the ship is felt least by going down below as far as possible in the middle of the ship. In a swing the body is very near the centre of oscillation, so that ordinary swinging is very little preparation for the motion of a vessel. A swing to act properly as a preparation for a sea voyage should be constructed as in fig. 5, to imitate, in full size, the cross section of the ship, suspended at M ; arid now the varying effect of the motion can be experienced by taking up different positions on the deck, up the mast, and in the cabins, con structed in this swing. Sir W. Thomson proposes to find the axis of rotation of a ship and the angle through which the ship rolls by noting the direction of the plumb lines of two such plummets, suspended THE ELLIPTIC FUNCTIONS. 87 at two given points across the ship ; planes through the plum mets perpendicular to the plumb lines at the extreme end of a roll would intersect in K\ the horizontal plane through K would meet the median longitudinal plane of the ship in the axis G ; while the plane through K perpendicular to the median plane would meet it in L, whence GL, the length of the equivalent pendulum, and therefore the period of small oscillations could be inferred, as a check on this construction. Example. A rod AB, whose density varies in any manner, is swung in a vertical plane about a horizontal axis through A. Prove that the bending moment of the rod is a maximum at a point P, determined by the condition that the C.G. of the part PB is the centre of oscillation of the pendulum. 97. PROBLEM VI The Elastica or Lintearia. The Elastica is the name given to the curve assumed by a uniform elastic beam, wire, or spring, originally straight, when bent into a plane curve (fig. 9) by a stress composed of two equal opposite forces T, on the assumption that at a point P at a distance y from the line of the applied stress the bending moment Ty is equilibrated by a moment of resistance B/p, proportional to the curvature l//o ; and the constant B is called the flexural rigidity of the spring (Thomson and Tait, Natural Philosophy, 611). O M G x B Fig. 9. Then Ty = B/p, or y p = B/T= c 2 , suppose ; and by KirchhofFs Kinetic Analogue, the normal of the Elas tica performs pendulum oscillations on each side of a perpen dicular to the line of stress, as the point on the curve moves with a constant velocity. 88 ILLUSTRATIONS OF For, when the normal has turned through an angle 0, the curvature - = -=- = 9- p ds c 2 and by differentiation d*6 1 dy 1 . ~r~2 = ~9 7 ~? sm $> as 2 c 2 <is c 2 which agrees with the equation of pendulum motion d 2 0/dP = - n 2 siu ft if 8/c = nt. Corresponding with the oscillating pendulum we have the undulating Elastica, intersecting the line of stress at an angle a ; and thus, writing s/c for nt in 8, sin JO = K sn s/c, cos JO = dn s/c, sin = dy/ds = 2/c sn s/c dn s/c, so that y = 2c K en s/c, measuring s from the point A, at a maximum distance from the line of thrust ; and a graduated bow might thus be employed for giving mechanically the numerical values of the en function. In the nodal Elastica corresponding with the revolving pendulum, = 2 am S/CK, sin = 2 sn S/CK en S/CK = dy/ds ; SO that y = 2(c//c) dn S/CK. In the separating case, K = 1, and y = 2c sech s/c ; and JO = amh s/c, sin JO = tanh s/c, tan JO = sinh s/c, etc. In the undulating Elastica /y /Y ~ = cos = ^/(l - 4/c 2 sn 2 s/c dn%/c) = 1 - 2/c 2 sn 2 s/c ; and in the nodal Elastica = cos = ^(l - 4 sn 2 s/c cn 2 s/c) =1-2 sn 2 s/c ; so that x is given in terms of s by means of elliptic integrals of the second kind ( 77). A great simplification is introduced when K = K=^ M J 2- J the Elastica now cuts the line of thrust at right angles, and cos = cn 2 s/c = J2/ 2 /c 2 , which shows that this Elastica is the roulette of the centre of a rectangular hyperbola, rolling on the line of thrust. It is easily proved that in this curve the radius of curvature p is half the normal PG ; also that a chain can hang in this curve as a catenary, provided the linear density is proportional to (ncs/c) 3 ; this is left as an exercise for the student. THE ELLIPTIC FUNCTIONS. 89 Wheri /c = 0, the undulating Elastica corresponds with small oscillations of the pendulum, and the Elastica is ultimately coincident with the line of thrust, the ordinate y varying as sins/c or sinx/c; and then the length of the beam, TTC = Tr^/(B/T), is the extreme length at which the straight form of the beam begins to become unstable under the thrust T. The nodal Elastica becomes practically a circle when /c = 0, corresponding in KirchhofFs Kinetic Analogue to the practi cally uniform revolutions of a pendulum when the velocity is indefinitely increased. The Elastica is also called Bernoulli s Lintearia, being the cross section of a horizontal flexible watertight cylinder, when filled with water, the free surface of which lies in the line of thrust Ox] for if t denotes the constant circumferential tension. t/p=wy, the pressure of the water, or yp = t/w = c 2 . It is also the profile of the surface of water drawn up by Capillary Attraction between two parallel plates (Maxwell, Encyclopaedia Britannica, Capillary Action). The student may prove, as an exercise, as in 80, that if the wire is bent into a tortuous curve by balancing forces and couples at its ends, it will assume the form of a curve in a surface of revolution defined by an equation of the form (Proc. London Math. Society, vol. XVIII.) 98. PROBLEM VII. Sumner Lines on Mercators Chart. Sumner Lines, so called after Captain Sumner, of Boston, Massachusetts, are the projections on M creator s chart of small circles on a sphere ; if simultaneous observations are taken of the chronometer and of the altitude of the sun or a star, the observer knows that he must lie on a small circle having its pole where the Sun or star at that instant was in the zenith, and having an angular radius the complement of the observed altitude; and two such observations are em ployed in Sumner s Method for determining the ship s place. According as the observed altitude of the Sun or the star is greater or less than the declination, the small circle on the 90 ILLUSTRATIONS OF Earth does not or does enclose the polar axis; and the cor responding Suraner line will be a closed or open curve, whose equation may be thrown into the form cosh y/c = sec a cos x/c, ....................... (i.) or sinh y/c = tan /3 cos x/c ....................... (ii.) On Mercator s chart ( 16) the latitude and the longitude of a point whose coordinates are x, y may be written where Trc/180 is the length on the chart of a degree of longitude at the equator. These relations are obtained by noticing that the bearing by compass of two adjacent points on the chart will be the same as on the terrestrial sphere, if dy_ dO dx cos 6d(p and now, if x = C(f>, so as to make the meridians of longitude equidistant parallel straight lines, then dy/dO = c sec 6, y/c =/sec OdO, or ( 16) $ = amh y/c. Now let <5 denote the declination of the Sun or star, y the observed altitude, < the difference of longitude of the observer and of the object ; then in the spherical triangle SPZ S denoting the Sun or star, Z the zenith of the observer, and P the pole of the Earth s axis. Since cos 8Z= cos PS cos PZ+ sin PS sin P^cos SPZ, therefore sin a = sin S sin 6 + cos S cos cos 0, or cos S cos (j> = sin a sec 6 sin S tan = sin a cosh y/c sin S sinh y/c ; and according as a is greater or less than S, this is reducible to the form A cosh(y b)/c or Bsiiih(y 6)/c; and this again by a change of axes to the form of (i.) or (ii.). (Crelle, XL, Gudermann, on the Loxodrome ; Messenger of Mathematics, XVI. and XX., Sumner Lines.) Differentiating equation (i.) with respect to x, dy _ sec a sin x/c _ sec asm x/c dx ~ sinh y/c /v /(sec 2 a cos 2 x/c 1) ds tan a _ sin a dx x /(sec 2 a cosPx/c 1) x /(sin 2 a sin 2 x/c) THE ELLIPTIC FUNCTIONS. 91 so that, as in 3, 4, and 8, sin x/c = K sn s/c, cos x/c = dn. s/c, cosh y/c = sin a dn s/c, sinh y/c = tan a en s/c, the modular angle being a. This shows that s/c in the closed Sumner Line (i.) may be equated to nt in the oscillating pendulum, and then x/c will be half the angle made by the pendulum with the vertical ; also in the Sumner Line dx cos \[s = -=- = cn. s/c, or i/r = am s/c, the intrinsic equation ; and p = c sin a sec x/c. The differentiation of equation (ii.) gives in a similar manner _ dx so that x/c = am s/c, with mod. angle /3 ; and now, in the corresponding undulating Sumner Line, x/c is half the angle made with the vertical by a revolving pendulum, if we put s/c = Knt. Also -T- , l//c) by 29 ; so that ^ = am(/cs/c, l//c), the intrinsic equation ; and p = c cosec /3 sec x/c. Fig. 10. The second curve, by a shift of origin a distance JTTC to the right, becomes sinh y/c = tan j3 sin x/c, and then it cuts at right angles the first curve (fig. 10) cosh y/c = sec a cos x/c. 92 ILLUSTRATIONS OF For, differentiating these equations logarithmically, ,->ydy ,x coth^ -/- = cot-. c dx c , y dy x tanh -- -^- = tan - , c dx c and therefore the product of the ~ s is 1. In fact .putting sec a coth a, the curves are derivable as conjugate functions from the equation x + iy = c amh(a / + i/3). 99. PROBLEM VIII. Catenaries. " The catenary for a line density proportional to cosh s/a, where s is the length of the arc measured from the lowest point, is of the form tanh y/b = dn x/a, or dn x/b, according as a, the ratio of the tension in pounds to the density in Ib. per foot at the lowest point of the catenary is greater or less than b ; the Catenary of Uniform Strength being the curve in the separating case of a = b" The equation of the Catenary of Uniform Strength, in which the linear density or cross section is so arranged as to be proportional to the tension, is well known (Thomson and Tait, Natural Philosophy, 583) being evl b cos x/b = l, or e^/ & = sec x/b ; or as it may be written tanh \y\~b = tsn\^x/b. For if O-Q denotes the density in Ib. per foot, and cr 6 the tension in pounds at the lowest point A, cr the density and 0-6 the tension at any other point P, at a distance s from A, measured along the curve, the equations of equilibrium of APare o-b cos \js = CT O &, o-b sin -^ =Ja-ds. Thence a- = o- sec \}s, andycrc/s = a- b tan i/r ; so that <r = <r 6 SQC 2 \lsd\js/ds = <r sec \]s, or ds/d-fi = b sec i/r, s =f b sec \fsd\fs = b cosh ~ ^ec \{s = b cosh ~ 1 o-/o- , o a- = o- cosh s/b. THE ELLIPTIC FUNCTIONS. 93 We might therefore take a piece of uniform flexible and inextensible material, cut out from a plane piece by two catenaries, or modified catenaries, say y/c= cosh x/b, and hang it up in a catenary of equal strength. Also x =/cos i/rcZs ==/*bd\lr = 61^, y =Jsiu \fsds =jb tan ijsdty = b log sec i/r ; so that y/b = log sec x/b, or e y/b = sec x/b t the equation of the Catenary of Uniform Strength. But now suppose two supports at the same level to be made to approach or recede from each other ; the piece of cloth or the chain will hang in a different catenary. Denoting by <r ct the tension in pounds at the lowest point A, and by t the tension at P, then t cos \fs = ar Q a, ts m\fs =f&ds = or b sinh s/b ; dy b . , s so that p or -~- = tan \^ = - smh y , the intrinsic equation of the curve. or v- aMp an elliptic integral, of the form (10), p. 33; and putting p = tan d\[r _ //cos 2 \/r sin 2 \/A ~- In the separating case, a = 6; and then x = b\fs, as in the Catenary of Uniform Strength ; the greatest possible span of a catenary of given material is therefore 7 rb = 7TT/ iv, where T denotes the tenacity of the material, in pounds per sq. foot, and w the density or heaviness, in Ib. per cubic foot. But with a > b, ^ *)> where K = b/a; so that JTT -f \/r = am x/b, A dy en x/b and -- = i^u\h-= --- L C en x/b sn x/b 7 f K 2 sux/b snaj/6 7 =/~ 2 n J -dx = /- I J sn 2 #/6 J l -- --- dx sn x/b x/b sn I dn 2 x/b or tan h y/b = d n x/b. ILLUSTRATIONS OF With a < b, so that \r = am x/a, -, cfo/ . sn x/a and -^=tanr= -- 4- j -- -, ewe en 05/a /sn #/a en oj/a 7 _ /*/c 2 sn cc/a en a/a - 1) - / -- Ct/X == / i - n - / - A> - cu z x/a J dtfx/a-K 2 o a or tanh y/b = i --- r = dn(^T aj/a), dnx/a by 57 ; so that by a change of origin, taking the axis of y in a vertical asymptote of the curve, its equation may be written tanh y/b = dn x/a. (Compare Cayley, on A Torse depending on Elliptic Func tions, Q. J. M., XIV., p. 241.) 100. In the catenary formed by an elastic rope or flexible wire, obeying Hooke s Law " ut tensio sic vis," we may still have p = sinh u ; but u is no longer proportional to the arc s. We use o- to denote the uniform density of the rope when unstretched, and s to denote the length of rope which stretches in AP to length s, a- Q b denotes as before the tension in pounds of the rope at the lowest point A, and CT O C is used to denote the modulus of elasticity of the rope in pounds ; so that, by Hooke s law, 1 = 1 -| -- . as Q o- c Then, as before, for the equilibrium of AP, t cos i/r = (r a, t sin i/r =fo-ds = or s Q) so that p = -^- -f = sinh u, dx o if we put s = a sinh u ; and then t = cr ox /(a 2 + s 2 ) = cr a cosh u. t \ds . a 2 m , ds A $ \ds Q , a 58 , o Inen ^ =1J ^_y = acO shuH cosh-u, /-y/j/ \ -_ /7//Vo/ /* Ct/ Cv \ O^n^ / t-v tv O and 7 - = ^/( 1 + p 2 ) = cosh u, dx THE ELLIPTIC FUNCTIONS. 95 ,1 dx a 2 , so that -= = cH cosh u, du c -z- = a sinh u-\ cosh u sinh u. du c Integrating, putting a/c = h, s/a = sinh u + %h(u + cosh u sinh u), x/a = u + h sinh u, 2//a = cosh u + J/i sinh%. For the corresponding points on the rope, when it is supposed inextensible, putting c = co , and h = 0, s /a = sinh u, xja = u, y /a = cosh u, giving an ordinary catenary ; so that the tangents are parallel at corresponding points of the catenaries of the elastic and of the inextensible rope. The terms depending on h, considered separately, define an ordinary parabola ; so that the catenary formed by an elastic rope is something intermediate to a parabola and a common catenary. 101. PROBLEM IX. Geodesies. " Investigation of the geodesies on the Catenoid, the surface formed by the revolution of a catenary round its directrix, and on the Helicoid, into which it can be developed ; also of the geodesies on the Unduloid and Nodoid, the capillary surfaces of revolution, of which the meridian curves are the roulette of the focus of a conic section, an ellipse or hyperbola, rolling upon the axis of revolution." The simplest mode of determining a geodesic on a surface of revolution is to treat it as the path of a particle moving under no forces on the surface, considered as smooth, so that ds/dt is constant ; and then, since the reaction of the surface passes through the axis, r z dO/dt is constant ; and therefore = 6, a constant, ds r and denoting the polar coordinates of any point of the projection on a plane perpendicular to the axis Ox ; and thus ds- dx 2 . dr 2 . r 4 96 ILLUSTRATIONS OF In the catenoid r/a = cosh x/a, so that dx a and therefore, in the geodesic, r 2_ a 2 dr 2 .dr 2 r* + ~ dO* b 2 We must distinguish the two cases according as b 2 ^ a 2 . When b 2 >a 2 , then r 2 >b 2 ; the geodesic osculates the circular cross section of radius b ; and we have r sn 6 fr, with K = a/b, as the polar equation of the projection of the geodesic. When b 2 < a 2 , then r 2 > a 2 ; the geodesic crosses the circular section of minimum radius a; and supposing it cuts the meridian here at an angle a,b = a sin a ; and now r sn($//c) = a, the modular angle being a. In the separating case, b = a and K = 1 ; and then sn(9 = tanh 6; so that r tanh = a is now the polar equation of the projection of the geodesic, a curve having r = a as an asymptotic circle. Generally in any geodesic on a surface of revolution, which cuts the meridian curve at a distance r from the axis at an angle v, sin y = r-- = ; A da r so that sin ^ varies inversely as r. 102. Now suppose the catenoid is divided along a meridian curve AP, and again along the smallest circular section A A , and that this section AA is drawn out into a straight line, of length 2-7TC& ; the rest of the surface, if flexible and inextensible, will assume the form of a Helicoid, or uniform screw surface of pitch a, such that its equation is z = a$, taking the axis of z along the axis of the surface, and p, the polar coordinates of the projection of a point on a plane per pendicular to the axis ; and AP will become a generating line of the Helicoid ; this is proved geometrically, by noticing that the length of the helix PP f on the Helicoid is equal to the length of the circle PP f on the Catenoid. THE ELLIPTIC FUNCTIONS. 97 The surface being inextensible, and a circular cross section of the Catenoid becoming a helix on the Helicoid, it follows that r 2 d6* = p-d<f + dz 2 = ( P 2 + a 2 )d<j> 2 ; and since r 2 = p 2 + a 2 , therefore 6 = <. Fig. II. Therefore the equation of the projection of a geodesic on the helicoid is either of the forms or (p 2 + a 2 )sn 2 = b 2 = a 2 /* 2 , _ a dn The Catenoid is the surface of revolution formed by a capillary soap bubble film, when the pressure of the air is the same on both sides of the film. The surface is easily formed practically by dipping a circular wire into soapy water and raising it vertically ; and it is evident from mechanical con siderations that the surface is a minimum surface ( 31). The Helicoid, into which the Catenoid can be deformed, can be produced in the same manner by a film between two coaxial helical wires of the same pitch (C. V. Boys, Soap Bubbles). G.E.F. 98 ILLUSTRATIONS OF These surfaces are particular cases of Scherk s minimum surface, whose equation is ,y ,aj(x* + y*-b 2 ) ,J(x* + y 2 -b*) s^atan-^ + ataji-V^TK-T 2 . 2\ + b tanh " // *T - 2 , 2 > # &vO 2 + 2/ + ) ^/(a; 2 + y + a ) or reducing to the Catenoid when a = 0, and to the Helicoid when 6 = 0. The verification in the manner of 32 is left as an exercise for the student. 103. The meridian curve of the Catenoid is the roulette AP of the focus of a parabola aG, the pressure of the air being the same on both sides of the film (fig. 12). But when the pressure of the air inside the film is increased or diminished, we find that the surface of revolution formed by the capillary film has as meridian curve BP or OP, the roulette of the focus of an ellipse or hyperbola, the first surface being called the Unduloid and the second the Nodoid. (Maxwell, Capillary Attraction, Encyclopaedia Britannica.) Denoting by y, y f the perpendiculars from the foci P, P on the axis Ox on which the conic rolls, then in the Unduloid BP, generated by the focus P of a rolling ellipse bQ, y + y = (PQ+ QP )COS ^ = 2a COS ^, and yy =W; so that 6 2 + y 2 = 2ay cos \js. If in the meridian curve BP of the Unduloid, we denote the radius of curvature by /o, and the normal PG by n, then, since b 2 + y 2 = 2ay cos ^ = 2ay*/n, 1 6 2 1 therefore - ^ 9 + s~ ; n 2a 2 2a and since cos \fs = -- + -, lay 2a differentiating, . d\!s ( b* 1 \dy sin y-f = ( - - - o - ) -/-, as \2a / ?/ 2 2a/as THE ELLIPTIC FUNCTIONS. 99 Fig. 12. or so that __ = _. n p a Then, if p denotes the excess over the atmospheric pressure of the air inside a capillary film, in the shape of an Unduloid, and t the tension of the film, p = t(-} = L: \n p/ a so that, if inside a Catenoid, the pressure is increased, the surface is changed into an Unduloid. If the pressure is slightly diminished by p, the surface be comes a portion of a Nodoid CP ; for now and in the meridian curve CP of the Nodoid, the roulette of the focus P of a hyperbola cR with foci P and P", y" y = (P"R RP)cos \fs = 2a cos \js, and yy = b 2 ; so that 6 2 y 2 = 2ay cos i/r = 2ay 2 /n : l = 6 2 1 n~2ay 2 2a and 100 ILLUSTRATIONS OF In the geodesic on the Unduloid, y 2 dO/ds = a sin y. supposing the geodesic cuts the meridian curve at an angle y at its maximum distance a from the axis; also a = a(l+6), and the minimum distance /3 = a(l e), so that a/5 = 6 2 , a-h/3 = 2a; and y lies between a and /3. Now, in the projection of the geodesic on a plane perpen dicular to Ox, writing r for y, so that tsm\[^ = dy/dx ds 2 dx 2 dr 2 dr 2 r 4 or 02 a 2 sm 2 y and ? cos \/s = (b 2 -\- r 2 )/2a ; so that = L ffl+^m ^ 1 4a 2 Aa 2 sin 2 leading to integrals of the form (72) and (73), p. 52. We suppose first that /3 > a sin y, so that the geodesic crosses the minimum section of the surface, and therefore all the sections if produced ; and now with a > r > /3 > a sin y, we have, according to equation (72), ~ J(a: 2 -r 2 .r 2 -/3 2 .r 2 -a 2 sm 2 7 ) 1 _cn 2 m^_ L sn 2 m0 r 2 a 2 /3 2 " Secondly, if a > r > a sin y > /3, then the geodesic osculates the circle of radius a sin y, and is limited by the convex part of the surface between two such circles ; and the equation of the projection of the geodesic is obtained from the above merely by interchanging a sin y and /3. In the separating case a sin y = /3 ; and then K = 1, m = tan Jy ; and the polar equation of the projection of the geodesic is 1 = sech 2 mfl tanh 2 mfl r 2 rf~ ~p*~~ a curve having an asymptotic circle y = /3. The formulas are similar for the geodesies on the Nodoid. THE ELLIPTIC FUNCTIONS : 104. Eulers Equations resumed. Poinsot s Geometrical Representation of the Motion of a Body under No Forces. We now resume these equations of motion, of which the solution by elliptic functions has been indicated in 32. By the Principle of the Conservation of Angular Momentum (Routh, Rigid Dynamics, Chap. IX.) the axis OC of the re sultant angular momentum G will be fixed in space ; and the direction cosines of this axis with respect to the principal axes of the body being Ap/G, Bq/G, Cr/G, the component angular velocity about OC will be Or 6r where, as before, T denotes twice the kinetic energy of the body. It is convenient to denote this component of angular velocity about OC by a single letter, say /m; and also to replace G and T by Z> M and D/x 2 , making T/G = ^ and G 2 /T= D; and then D will be a constant quantity, of the same dimensions as A, B, C. If / denotes the moment of inertia about the instantaneous axis of rotation OP, and if OP denotes the vector of the momental ellipsoid at 0, then /varies as OP~ 2 , so that we may put I=Dh 2 /OP 2 , where h is a new constant length. Now, if o> denotes the resultant angular velocity about OP, T-Jo) 2 , or D^ = DhVjOP\ so that the angular velocity CD varies as OP : and IUL CD p q r The direction cosines of the normal of the momental ellipsoid at P being proportional to Ax, By, Cz, or Ap, Bq, Cr, are therefore Ap/G, Bq/G, Cr/G ; so that OC, the axis of G, is perpendicular to the tangent plane at P ; and if OC meets this tangent plane in C, it follows that OC = h, so that the tangent plane at P is a fixed plane ; and during the motion the momental ellipsoid rolls on this fixed plane, called the in variable plane, with angular velocity proportional to OP. The curve traced out by the point of contact P on the momental ellipsoid is called the polhode, and the curve traced out by P on the invariable plane is called the herpolhodc ; ILLUSTRATIONS OF these names are due to Poinsot, as well as this geometrical representation of the motion. (Theorie nouvelle de la rotation des corps, Paris, 1852.) The equation of the momental ellipsoid may now be written while Ax/Dli, By/Dh, CzjDh are the direction cosines of the invariable line 00 ; so that AW + B*y* + C V = D% 2 . The polhode is therefore the curve of intersection of these two coaxial quadric surfaces, and therefore lies on the cone called the polhode cone ; and the projections of the polhode on the principal planes are therefore (A - B)By* + (A-C) Gz l = (A- D)Dh z , .... 105. Denoting by v the component angular velocity of the body about the axis OH, where OH is equal and parallel to CP t and, by solution of these equations, A-B.A-C . or = i/ 2 + (l - -g) (I - Q) yu 2 = ^ - v,?, suppose ; B-C.B-A /- I>\- D C-A.C-B 2 / D\(. D -^^ 1 " and in these equations we may replace p, q, r, o>, /x, v by ^, T/, 0, OP, /t, p, respectively, where p 2 = OP 2 h 2 . Example. Prove that and simplify THE ELLIPTIC FUNCTIONS. 103 106. On the supposition that AT>BT> G 2 > CT, orA>B>D>C, r never vanishes, and the polhode encloses the principal axis (?; but p and q alternately vanish, so that i/ 2 oscillates in value If we put v - 2 = (~ - l){(l - ^)cos 2 6> + (l - 5 /r \U \ & * o/ then A- Or* = We now find, on substituting in one of Euler s equations, -D . , d 2 n 9 (A-B)(D-C) . and ~- = - D^- . -- - sm 6 cos 6, the solution of which is of the form, as before in IS and 32, = a,m(nt, K ), ,A-D.B-C A-B.D-C n ~ = D ^- and K - = the anharmonic ratio of A, B, D, C ; while giving ( 32) 107. Quadrantal Oscillations. The oscillations given by a differential equation of the form are called quadrantal oscillations (Thomson and Tait, Natural Philosophy, 322), the system having two positions of stable 104 ILLUSTRATIONS OF equilibrium given by = and = TT, and two unstable posir tions in the remaining quadrants, given by 6 = |-TT ; for instance, an elongated piece of soft iron in a uniform magnetic field, or an elliptic cylinder moveable about its axis in a cur rent of liquid performs quadrantal oscillations. (Q. J. M. t xvi.) When the system performs complete revolutions, the solu tion is ( 18) = &m(mt/K, K) ; but if it oscillates about the positions of stable equilibrium, given by 6 = 0, the solution is ( 29) 0= am(m, l//c), or cos 6 = dn(mt/K, K), sin 9 = Kfm(mt/K, AC), where K is less than unity. The second solution will apply to the second state of motion in 32, where AT> G 2 > BT> CT, or A > D> B > G, and where p never vanishes, and the polhode encloses the principal axis A . 108. Differentiating the equations of 105 with respect to t, du_ dv_A-B.A-G dp^B-G.B-A dg_G-A.G-B dr <a dt~ v dt~ BG P dt~ CA q dt~ AB ~ r di _B-C.C-A.A-B ABC ~ PqT or = - 4. . o, a 2 - a, 2 . W 2 - a> 2 . so that co 2 and t/ 2 are elliptic functions of t, of the form given by equation (15), p. 36. But, on reference to equation (A), p. 43, we see that if e a , e b , e c denote the roots of 4s 3 # 2 s # 3 = ; so that on comparison we may make proportional to $u e a , ^u e^ $)u e c ; or, symmetrically, we can put C T * = - m\A - B)($u - e c ) ; where the factor -m 2 is introduced for the sake of homogeneity, THE ELLIPTIC FUNCTIONS. 105 m being of the dimensions of an angular velocity, such as p, q, r, CD, [*, v ; and now, on substitution in Euler s equations, du* B-C.C-A.A-B (B-C C-A A- ~dt* = -ABC- - m =(^r+-jr+-c suppose; so that u = & constant nt. 109. As in 32, we take A > B> C- and then (i.) when AT>BT>G 2 >CT, or A>B>D>G, r never vanishes, and we must take e c > e a > pu > e b ; so that ^ = e c , e 2 = e a> e 3 = e b ; (ii.) when AT> G 2 >BT> CT, or A >D>B>C, p never vanishes ; and then and we must take e 1 = e a , e, = e c , e. = e b . Since pu oscillates between e. 2 and e 3 , and is taken initially equal to e s , we find, on reference to equation (42), p. 45, that we must put u = 2^ + o> 3 nt, so that the constant of integration for u in 108 is 2^ + 0)3. Now, at the cost of symmetry, to get rid of the imaginary o> 3 , and to make the argument of the elliptic functions a real quantity nt, equation (42), expressed in the direct notation, gves B and e b always replaces e 3 , while e a replaces e v e c replaces e 2 , or vice versa, according as the polhode encloses A or (7. 110. For the determination of e at e b , e c , we have the equations *a+ e b + e c = 0, (B-C)e a + (C-A)e b + (A - B)e c = T/m 2 = D^fm 2 , A(B- 0)e a + B(C-A)e b + C(A - B)e c = G*/m* = D^/m*, whence A T- G 2 = m 2 (C -A)(A- B)(e b - e c ), BT- G* = m *(A -B}(B- C)(e c - fl ), 106 ILLUSTRATIONS OF A-D 6 *~ B-D a m 2 A-B. B-V = D/x 2 C-D m*~B-C.C-A so that e c e a is taken positive or negative, according as BTG 2 or B D is positive or negative; while e b e c and e b e a are always negative, as explained above. Also (ea eb)-(< c-ta) = 3sa, > whence the values of e a , e b , e c . Then </ 2 = {(e 6 - e c ) 2 + (e e - e a )* + (e a - e fr ) 2 } can be found ; and the discriminant ( 53) = r_^2 3 _- ~ ~~ 111. We have supposed no forces to act; but the case in which the impressed couple is always parallel and proportional to the resultant angular momentum leads to equations which can be solved in a similar manner ; in this way we imitate the motion of a body, like the Earth, which is cooling and con tracting uniformly. Now, the component impressed couples about the principal axes being of the form \Ap, \Bq, XCr, A(dp/dt)-(B-C)qr = \Ap, ..., which, on putting p = e~ Xt p , and \t / = l e~ Xt , reduce to so that p , q f , r are the same functions of , which p, q, r would be of t, in the case where no forces act. In the case of the cooling and contracting body, we put A=e~ Xt A , B = e~ Xt B , C=e~ Xt C Q - ) and the equations become which are solved as before ; and Poinsot s geometrical repre sentation of the motion still holds, with slight modification. THE ELLIPTIC FUNCTIONS. 107 A similar procedure will solve the following theorem : " A rigid body is moving under the action of a force whose direction and magnitude are constant, always passing through the centre of inertia (e.g. gravity), and of an absolutely con stant couple. " If p, q, T denote the component angular velocities about the principal axes at the centre of inertia, and if u, v, w denote the compound velocities of the centre of inertia along the principal axes at the time t ; then the determination of p/t, q/t, r/t, ujt, v!t, w/t, in terms of J 2 is the same as that of p, q, r, u, v, iv, in terms of t, when no forces act; t being reckoned from the commence ment of the motion." (W. Burnside, Math. Tripos, 1881.) 112. To obtain the equation of the herpolhode, we notice that during the motion the polhode cone, fixed in the body, rolls on the herpolhode cone, fixed in space, being the common vertex ; corresponding areas of these cones are therefore equal, as also their projections on any fixed plane, for instance the invariable plane. Therefore if p, (p denote with respect to C the polar co ordinates of P on the herpolhode, dz d\ B dx dz Cz d dx Since *=0=?=fi p q r v therefore , dt h A dz d V ufA-B C-A , .1 ABC D.C-D ABC which, combined with the value of di?/dt or dp 2 /dt of g 108, Pa" ~ P Pb~ - p~ will determine the equation of the herpolhode. 108 ILLUSTRATIONS OF 113. Using Weierstrass s functions of 108, M 2 1 mJ M _ . C-A . J.- u 2 xl a with Of = - ^-(7 C-J. A-B - i i and then i w ""^ SSB1 - L j^- 1 (positive), /A 2 /l ^/^> !\ ^-^ = l--l. (positive), and, since 6 X (or e e ) >$v> e 2 (or <? a ), we must, by (39), 54, where t is a proper fraction, take v = Therefore ^ at or and, integrating, and we are thus introduced to a new integral, called an elliptic integral of the third kind. The cone described in the body by OH ( 105) is called by Poinsot the rolling and sliding cone ; during the motion this cone rolls on an invariable plane through 0, while at the same time this plane turns with constant angular velocity JUL about OC ; so that, if p, $ denote with respect to the polar co ordinates of H on this plane, THE ELLIPTIC FUNCTIONS. 109 114. With the notation of the elliptic functions of Jacobi, as in 106, p~ DD-C DD-C DA-D A-D.D-C D.A-B.D-C ^ -AC~ ABC which can be thrown into the form A DA-B on putting /c 2 sn 2 a = - -^ DB-C C B-D A B-D - a = BD=C> Cn " a= ~B D=C> dna = B A=D With e a = e z , e b = e 3 , e c = e lt and v = coj + t w s , then by (32), p. 44, and e b B AD so that a = K+t iK r . B-D _ B l-K 2 s i en a dn a n sna and, waiting u for nt, i sn a dn i en a dn a . /"/c sn a en a dn a sn 2 w sn a o . /"/c sn a en a u^/ y 5 ^y 1 /rsn-a 7 du, the last term an elliptic integral of the third kind, in the form employed by Jacobi. On putting sni6 = sin 0, and sn a = sin a, /c 2 sn 2 a= m, then ,.cosaAa sn o the third elliptic integral, as employed by Legendre ; the further discussion of this integral must be reserved for a subsequent chapter. 1 10 ILLUSTRATIONS OF EXAMPLES. 1. Prove that, if the excentric anomaly in an undisturbed planetary orbit of excentricity e is represented by 2 am(u, e), the mean anomaly is 2 am 2. Prove that the envelope of the straight line rays K Z X sn u + (en u + K dn u)y = K sn u(dn u + /c en u) where u is the variable parameter, is the curve the caustic of parallel rays, after refraction at a circle, of refractive index l//c ; and find the order of this curve. (Cayley, Phil. Trans., 1857, " Caustics.") 3. Prove that a portion of a flexible inextensible spherical surface of radius a, bounded by two meridians (a lune, or gore of a spherical balloon) can be bent into the surface of revolu tion given by , /c; 0, $ denoting the latitude and longitude of the point on the sphere. Explain the geometrical theory, distinguishing the cases of K <1, and K >1. 4. Denoting by o> the solid angle subtended by a circle of radius a at a point whose cylindrical coordinates are r, z with respect to the axis of the circle, prove that dco _ az K 3 da~ 2arl K - Show how to determine the illumination at any point of the surface of the water at the bottom of a deep well, due to the light from the sky. THE ELLIPTIC FUNCTIONS. HI 5. A uniform circular wire, charged with e coulombs, is presented symmetrically to a fixed insulated sphere of radius a centimetres, so that every point of the wire is at a distance / cm from the centre of the sphere, the radius of the wire sub tending an angle a at the centre of the sphere. Prove that the electricity, in coulombs per cm 2 , induced at a point of the sphere whose angular distance from the axis of symmetry is 0, is given by E .,_ Wsinasinfl /2 _ a 2 - 2a/cos(fl - a) +/ 2 ~ a 2 - 2a/cos(0 + a) +/ 2 ~ a 2 - 2a/cos(0 + a) +f* 6. Prove that if this sphere and wire gravitate to each other, and if the wire is free to turn about a fixed diameter perpen dicular to the line joining the centres, the wire will be in stable equilibrium when its plane passes through the centre of the sphere ; and prove that the oscillations of the wire due to the gravitation will synchronize with a pendulum of length ) CJ cm where b denotes the radius of the wire, c the distance between the centres of the sphere and wire in cm, M the weight of the sphere in g, C the gravitation constant ; and JP= lj-X= J K (1 +O jr {(1 + K -)^K], where K~ = bc/(b + c) 2 . Determine the position of stable equilibrium and the length of the equivalent pendulum, when the attraction is changed to repulsion. 7. Two uniform concentric circular wires of radii b and c cm, weighing M and M g, are freely moveable about a common fixed diameter. Prove that in consequence of their gravitation, the oscillations will synchronize with a pendulum of length Cm where F and K have the same values as before. CHAPTER IV. THE ADDITION THEOREM FOR ELLIPTIC FUNCTIONS. 115. So far we have considered the elliptic functions of a single argument u ; but now we have to determine the for mulas which give the elliptic functions of the sum or difference, uv, of two arguments u and v, in terms of the elliptic functions of u and v ; and thence generally the formulas for the elliptic functions of the sum of any number of arguments u-\-v+w+...\ and the formulas for the duplication, triplication, etc., of the argument. The Addition Theorem for Circular and Hyperbolic Functions. The analogous formulas in Trigonometry for the Circular Functions are well known, namely, sin(u v) = sin u cos v cos u sin v, cos(t6 v) = cos u cos v + sin u sin v ; or, as they may be written, siu(uv ) = sin u sin i> sin% sin v, cos(u v) = cos u cos v + cos u cos v ; the accents denoting differentiation ; and to these*may be added . - 1 + tan u tan v these formulas constituting the Addition Theorem for the Circular Functions. For the Hyperbolic Functions, the formulas are cosh(u v) = cosh u cosh v sinh u sinh v, sinh(u v) = sinh u cosh v cosh u sinh v ; 112 ADDITION THEOREM FOR ELLIPTIC FUNCTIONS. H3 or, as they may be written, cosh( u v) = cosh u cosh v cosh u cosh v, sinh(i6 r) = sinh u sinhVsinh u sinh v ; and to these may be added ,, v tanh it tanh -v tanh(u v) = -, ; 1 tanh u tanh v constituting the Addition Theorem for the Hyperbolic Func tions. 116. The Addition Theorem for the Elliptic Functions. For the Elliptic Functions the analogous formulas of the Addition Theorem are found to be sn(uv) = (sn u sn / t? sn w- sn v)/D, cn(u v) = ( en u en v + cn u cn v)/D, dn(u v) = (dn u dn v /c~ 2 dn% dnv)/D, where D=l /c 2 sn%sn 2 v ; or, performing the differentiations, and dropping the double signs, sn u en v dn v + en u dn u sn v /n cn u cn v sn u dn u sn v dn v 1 K 2 sri 2 u sn 2 y dn it dn v /c 2 sn it cn u sn v cn v /ox (3) Putting /c = 0, we obtain the formulas for the Circular Functions, sin(u + v) and cos(w-f-i ), the denominator D re ducing to unity. Putting /c = l, remembering that then (16) snu becomes tanhu, cnu or dnu becomes sech u, we obtain from (1) tanh u sech 2 v + sech 2 ^ tanh v , 9 , 9 1 tanh 2 w, tanh 2 y _ tanh u( 1 tanh 2 1?) + ( 1 tanh %) tanh v _ tanh u + tanh v 1 tanh 2 u tanh 2 i; 1 + tanh u tanh v as before; with the corresponding formula for sech(i6-f-v) or cosh(w + r), the formulas for the Hyperbolic Functions. 117. To establish these formulas of the Addition Theorem for Elliptic Functions, let us employ the geometry invented by Jacobi (Crelle, Band 3 ; Gesammelte Werke, I, p. 279), at the same time interpreting the geometry in connexion with Pendulum Motion. G.E.F. H 114 THE ADDITION THEOEEM To do this, let us suppose that P f would be the position of P in fig. 2 at the time t, if it had started r seconds later, and puU- T = f; then (6) AN = AD stfntf, N D = AD cnV, N E=AE drfnt , etc. ; and we shall prove that PP f touches a fixed circle through B and B during the motion (fig. 13). For suppose that, in the small element of time dt, P has moved to an adjacent point p and P to p ; and let PP , pp intersect in R, so that R is ultimately the point of contact on the envelope of PP . Then since, by a property of the circle, PP cuts the circle APP at equal angles at P and P , PR _, Pp _ velocity of P _ I ND RP ~ Pp ~ velocity of P ~ V Ntf Now describe a circle with centre o on AE, passing through B and B , and touching PP at a point which we shall denote by R ; then = OB* + Oo*-Wo.ON- Bo* = Oo(OD+Do + oO- Similarly, so that PR PR FOR ELLIPTIC FUNCTIONS. 115 and therefore R and R coincide ; and we have thus verified that PP touches at R the circle oR (using the notation oR to mean a circle of centre o, and radius oR). Putting Oo = a, and denoting the angles A OP, A OP by 9, , and ADQ, ADQ by 0, ^, then PR 2 = 2a . ND = 4aR cos ^ = 4aZ/c 2 cos 2 0, so that P R+RP = 2 /v /(a while P P = 2Z sin J(0 - ), and therefore sin J(0 / ) = ^/(a/Z)K(cos i/r-f- cos 0). Putting nt = u, nt = v, nr = uv = iv; then since (8) , sin J$ = /c sin = /c sn u, cos J$ = dn u ; sin JO = /csini/r = (0 ) snudnv V<z j = 6 , a constant. I /f(CQS \lr + COS 0) Putting t = 0, v = 0, and therefore u = n-T = w, we find Va_ snw _lcmv_ 11cn.w. I 1 + cnw smu \l + cn^ so that 1 cn(u v)_suu dn v dn u sn v _ en v en ?& 1 + cn(i6 v) en v + en u sn u dn v + dn u sn u one form of the Addition Theorem, which by algebraical trans formation can be reduced to one of the preceding forms of 116. 118. Representing, as in 31, snu by s v cnu by c lt dnu by d lt and the corresponding functions of v by s 2 , c 2 ,.c? 2 ; then 1 + cn(u v) so that l-cn(u-t.j = fe-c i ; .-v) (c. V x or cn(u v) = and changing the sign of v, another form of the Addition Equation. A . 1 cn(u v) fs,d 9 Again 4 = -" 1+cnO-v) t A_ 2 - " and, adding numerators and denominators (componendo), THE ADDITION THEOREM . 2 (C (u V) = -9- -L K 8^ 82 the usual form (2) of the Addition Theorem for the en function. But, subtracting numerators and denominators (dividendo), Cn(u V) ~~ nT 1 -~|- _l_ s /_\2 + K 2, *+lf> and another form can be easily established in the same way, (Glaisher, Messenger of Mathematics, vol. x., p. 106 ; M. M. U. Wilkinson, Proc. London Math. Soc., vol. xiii., p. 109; Woolsey Johnson, Messenger of Mathematics, vol. xi., p. 138.) 119. Expressed again in Legendre s trigonometrical form, with = am u, i//- = am v, y = am(u v), ja _ 1 cos y _ sin <^> A^ sin V ^ sin y cos i//- + cos V _ 1 + cos y _ sin A\/^ + sin a sin y cos \^ cos Therefore, eliminating Ax//-, = 2 cos + 2 cos \/r cos y, or cos = cos \^ cos y sin -0- sin yA0. Expressed in Jacobi s notation, since u = cn.(v + iv) = cii vciiw Changing v + ^v into u v, this becomes cn(u v) = en u en v + sn u sn v dn(i6 v), or cos y = cos (p cos \^ + sin <^> sin \k-Ay. Conversely, these relations, treating y as constant, lead to the differential relations du dv = 0, or cZ0/A0-^/A^ = 0, or l - 2 sin 2 - - d 2 l - K 2 sin 2 < = 0. FOR ELLIPTIC FUNCTIONS. 117 Writing x for sin <p sin \/s, y for cos (p cos \fs, and m for Ay, then cos y = x /(m 2 /c 2 )//c ( 17); and the integral relation becomes y 4- mx = ^/(m 2 /c /2 )//c, leading to the differential equation, of Clairaut s form, y - xp = *J(j? - K Z )/K, denoting dy/dx by p ; this is the form of the differential equation when we change to these new variables x and y. 120. We have begun in 117 by supposing the points P and P to oscillate on a circle with velocity due to the level of the horizontal line BDB f , cutting the circle in B and B (figs. 2, 13); but if they are performing complete revolutions with velocity due to the level of a horizontal line BB through D not cutting the circle, but lying above it (figs. 3, 14), a similar proof will show that PP touches a fixed circle having with the circle PP the common radical axis BB , the two circles not inter secting ; and the Landen point L ( 28) will be a limiting point of these two circles. But this motion of P and P in fig. 14 is imitated by the circulating motion of Q and Q on the circle AQ in fig. 13 ; so that QQ touches at T a fixed circle, centre c ; and the hori zontal line through E is the common radical axis of this circle and the circle CQ, the Landen point L being a limiting point : and thus the Addition Theorem for Elliptic Functions can be deduced from the motion of P and P in fig. 14, or of Q and Q in fig. 13, as given by Durege, Elliptische Functionen, X. For if in fig. 1 4 a circle is drawn with centre o and radius oR, such that BDB (fig. 3) is the common radical axis of this circle and of the circle AP, then, since the tangents to these circles from D are equal in length, and now, if the tangent to the inner circle at R cuts the outer circle in P and P , as in 117 ; and similarly RP 2 = 20o . ND ; so that PR _ /^ / J D_ velocity of P . RP V XJJ velocity^fP" and therefore PP will continue to touch the circle R, during the subsequent motion of P and P . 118 THE ADDITION THEOREM Similarly, in fig. 13, QQ during the motion touches a fixed circle, centre c and radius cT ; and putting Oc = c, We notice, on reference to 28, that LQ 2 = 2LC .EN=2LC.EA drfnt = 4l\I so that LQ = LA dn nt ; and therefore -~ = -^, or XT 7 bisects the angle QLQ in fig. 13; while LR bisects the angle PLP in fig. 14 ; we may state this theorem geometrically, " the segments of a tangent to one circle, cut off by another circle, subtend equal angles at a limiting point of the two circles." Then, with the notation of 117, QT+ TQ = 2 </(cQ(A^+ A0), and Q Q = 2R sin(0 - \j/) = 2 K H sin(0 - \/r) ; so that, in Legendre s trigonometrical form, Putting \/r = 0, then = y ; so that \^)_ /csiny 1 Ay /csiny VR /csin(0 + \^) K sin y 1 + Ay = : *- : * = or - , c A\^ A0 1 Ay /csiny the product of the two equations being unity. Conversely, the relation sin(0 + \fs} = (7( A^ + A0), where C is an arbitrary constant, leads to the differential relation 121. Taking the equations Ay _ /c 2 sin(</> sny ir < sny we find, on eliminating sin 0, 2/c 2 cos sin i/r sin y = (1 + Ay) (A^ - A^) - ( 1 - Ay)(Ai/r + A^) A^ = AyAx/>" /c 2 cos ^> sin i/r sin y, or dn u = dn v dn it; /c 2 cn it sn v sn u ? , with it = FOR ELLIPTIC FUNCTIONS. 119 By eliminating cos 0, 2/c 2 sin cos \}s sin y = 2 A^ 2 AyA</>, Ai/r = A< Ay + /c 2 sin cos \/r sin y, or dn(u 10) = dn u dn 10 -f /c 2 sn % sn w cn(it w). Changing w into v, dn(u v) = dn u dn v -f /c 2 sn u sn v cn(i6 r), or Ay = A0 Ai/r + /c 2 sin sin i/r cos y. Writing a; for /c 2 sin sin i/^, ^/ for A^Ai//-, and 7)1, for cny, then y + mx = *J(ic 2 + /c 2 m 2 ), the integral relation of Clairaut s differential equation y - xp = *J( K 2 + K z p z ), which is therefore the transformation of when we change to these new variables x and y. Taking the two trigonometrical expressions from 119, 120, for the Addition Theorem, 1 cos y _ sin Ai//- sin 1//-A0 1 Ay _ K 2 sin(< i//-) siny cosi/r+cos< siny we obtain, by subtraction and reduction, Ay cos cos \[sA<p cos sin y sin <p + sin \js dn(u v) cn(i6 v) dn u en v en u dn v sn(u v) sn u -f sn v the form of the Addition Theorem given by J. J. Thomson (Messenger of Mathematics, vol. IX., p. 53). 122. With the notation of the elliptic functions, 1 + dn(u v) _ Ac(sn u en v + sn v en u} K sn(i// v) dn v dn u 1 du(u v) /c(sn u en v sn v en u) /c sn(it v) dn v + dn u Therefore, as before, with Glaisher s abbreviations, v) _ (cZ 2 cZ 1 )(s 1 c 2 - ~ 120 THE ADDITION THEOREM Similar algebraical reductions to those given above for cn(u v) will establish the formulas for du(u v) and dn(u+v), given by Glaisher (Messenger, X., p. 106), S l^-2 C 2 S 2^1 C 1 + /c V 2 c i c 2 1 - A V the last of form (3), 116. 123. The Duplication, Triplication, etc., Formulas. Putting v = u in formulas (1), (2), (3) of 116, and writing s, c, d for sn u, en u, dn u, we find 9 = Writing ^f, 0, D for sn2u, en 2u, dii 2u, we find 1+1) "" c/ 2 _ ~ Putting u=pr, then fif=l, 0-0, !) = /; and Again, in 67, , (1 +Qsn(^ /c)cn(u, /c) _ 1 +/ /l-dn(2u,/c) dn(u,/c) /c Vl + dn(2u,AcT and 2w=(l+X)v, X = (l -/)/(!+/), which is called Landeris second transformation. FOR ELLIPTIC FUNCTIONS. 121 Again, putting v = 2u, and making use of the above formulas, we shall find o n 3,,- 1 - 6* V + 4(1+ *Vs 6 - 3/cV 1 - sn 3u 1 + */l - 2s + 2/cV - /c 1 + sn 3u 1 - s\ 1 + 2s - 2/c 2 s 3 - /c l-jcsn3u_ 1 + K8/1 - 2/cS + 2/cs 3 - /cV\ 2 . 1 + K sn 3u 1 - #s \1 + 2/cs - 2/cs 3 - jc with similar expressions for en 3 it and dn 3tt, leading to + 2/c /2 c + 2/c 2 c 3 + * l-dn3u 1- l+dn3w, i_j_9i/r/3 9 t - / r7 I ** <-l/ -K Uj the algebraical work is left as an exercise for the student. 124. Poristic Polygons ofPoncelet, with respect to two Circles. Starting from the point A in fig. 13, and drawing the successive tangents AQ V Q^o, Q^Q^ to the inner circle, centre c, from the points Q v Q 2 , Q 3 , ... on the circle CQ ; or starting from A in fig. 14, and drawing the tangents AP V PJPv P 2 P B , ... to the inner circle, centre o, from P p P 2 , P 3 , . . . on the circle OP ; then, if we denote the first angle ADQ l or AEP 1 by am w, it follows from this construction that ADQ 2 = AEP = am 2^(;, ADQ 3 = AEP S = am 3w, . . . ; and we have thus a geometrical construction for the elliptic functions of the duplicated, triplicated, ... argument. When iv is an aliquot part, one 71 th , of the half period 27T, or T of the half period 2T seconds, then after n such operations the polygon AQ^^Q^, ... , or AP^^P^, ... , will close on itself at the starting point A ; and the preceding investigations show that during the subsequent motion of these points, the polygon formed by them will continue to be a closed polygon, inscribed in the circle CQ and circumscribed to the circle cT, or inscribed in the circle OP and circumscribed to the circle oR ; and thus we have a mechanical proof of Poncelet s Poristic Theorem for two circles, a problem discussed by Fuss, Steiner, Jacobi, Richelot, and Minding. (Cayley, Philosophical Magazine, 1853, 1854, 1861.) 122 THE ADDITION THEOKEM Let us consider the particular cases of w equal to , J, J, }, * ... of the half period 2K. (i.) When w = 2K, PP is horizontal in fig. 13; and P and P coincide in fig. 14. (ii.) When w = K, the circle oR in fig. 14 and the circle cT in fig. 13 shrink up into the limiting point L, Landen s point ( 28) ; and now any straight line through L will divide these circles OP or CQ into two parts described in equal times, ^T ; while in fig. 13 the line PP will touch the circle described with centre E through B, L, and B , subtending an angle 4a at ; and any arc PP will be described in time \T, half the time of describing BAB ; hence the following theorem " Two segments of circles are described on the under side of the same horizontal straight line, one subtending twice as many degrees at the centre as the other; if a particle oscillates on the lower segmental arc under gravity, any tangent to the upper arc will cut off from the lower an arc described in half the time of oscillation." (Maxwell, Math. Tripos, 1866.) As P is passing through A in fig. 15, P is instantaneously at rest at B or jB ; and AB, AB are obviously tangents at B and B to the circle BLB , drawn with centre E ; while PP is one side of a crossed quadrilateral, escribed to this circle BLB t and inscribed in the circle BAB . When the circle cT shrinks up into the limiting point L, then, as in 120, QL 2 = 2CL. EN, LQ 2 = 2CL. EN and since QL . LQ is constant in the circle CQ, therefore EN. EN is constant, and equal to LE 2 , the value it assumes when N and N pass each other at the point L. Since EN . EN = EL 2 = EB 2 , a circle can be drawn passing through N, N , and touching EB at B ; and the triangles ENB, EBN are therefore similar, so that ENB = EBN , EN B = EBN. (Landen, Phil. Trans., 1771, p. 308.) Translated into a theorem of elliptic functions, EN. EN = EA 2 du 2 udu 2 v, and EB 2 = K 2 . EA 2 , so that, as in (59), 57, dn u dn v = K, when u v = K, FOR ELLIPTIC FUNCTIONS. Otherwise, since ( 28) and therefore 123 = ALdiiv, QL.LQ = AL.LD, dn u dn v = LD/A L = K. A Fig. 15. The similarity of the triangles A QL, LDQ shows that and since ( 10) AQ = ADsn u, DQ = therefore, as in (57), 57, sn u = en v/dn. v or cd v, when u> = v Again, since DQ /DL = A Q/LQ, therefore DL sn u AL dn u dn u as in (58), 57, when v=-u K. 124 THE ADDITION THEOREM Conversely, if the straight line QLQ , passing through L, moves into the adjacent position qLq , then u <lQ _ Q^_ _ I EN _ velocity of Q q Q -LQ -^~EN ~ velocity of Q 7 if Q and Q move under gravity, or diluted gravity, on the circle CQ with velocity due to the level of E\ so that QLQ will continue to pass through L, and will divide the circle CQ into two parts described in the same time \T ( 28). If in fig. 13 we denote the radius of the circle cT by r, then y or am w denoting the angle ADQ-^ ; while, from 120, 1 Ay_ c . _R c, I+Ay""l > Cr y A , 2 4cR 2 (-R-c) 2 -r 2 and thence J = * = Again, if Z)g is drawn from D to touch the circle cT, and the angle ADq is denoted by y or am w , then , r cosy , cnw sin y = -^ = / r- L , or sn ^v = -= -- , ^L c Ay on 10 so that ( 57) 125. Poristic Triangles. (iii.) When w = \K or |-^T, triangles Q^Q^Qs can be inscribed in the circle CQ and circumscribed to the circle cT, while at the same time triangles P-,P 2 P 3 (or hexagons) can be inscribed in the circle OP and escribed to the circle oR (fig. 16). The well known relations of Trigonometry c 2 = R*- 2Rr, or a 2 = R 2 + 2Rr, where Cc = c, Oo = a, cT=r, oR = r, are now easily deduced. We may write these relations, more symmetrically, r r _ - ^ ~ r In fig. 16, and since cQg bisects the angle J\T|Qj-4., which is equal to y, therefore DcOi = J (TT y) ; and DcQ| = DQ^c, or D(^ = DC. Similarly AQ = Ac f so that Therefore sin y + cos y = 1 , or sn \K + CD.$K = I , r r + ~ FOR ELLIPTIC FUNCTIONS. 125 We shall employ this suffix notation for the points N, P, Q to signify points corresponding to aliquot parts of K. Corresponding to w = ^K, the circle oR becomes the circle through B, N^ R ; and nowP^AP* is a triangle escribed to this circle, and inscribed in the circle OP. For iv = ^K, the circle oR becomes the circle through B, N^ B ; and now we shall find that hexagons can be escribed to this circle, and inscribed in the circle OP. The tangents at P g , P* touch the circle BNJB , and the tangents at PI, Pj. touch the circle BNJ1 ; while AP%, AP< are the common tangents of the circles BNB f , BN 2 B . Denoting the sides of the triangle Q^Q^ by q v q z , g 3 , then But u v i// 2 , M 3 denoting the value of u corresponding to the points Q v Q 2 , Q B , and cZ 1? cZ 2 , d B denoting the corresponding values of dn u, then ( 120) (r7 2 so that a constant, a relation connecting d v d 2 , d s , when 126 THE ADDITION THEOREM 126. Poristic Quadrilaterals. (iv.) When w = ^K, quadrilaterals Q^Q^Q^Q^ can be inscribed in the circle CQ which are circumscribed to the circle cT, and now the corresponding relation is found to be ( -< v \Ji-J while TjjPg, T 2 T intersect at right angles in L, being the bisectors of the angles between Q^LQ^ Q?LQ 3 (fig. 17). This relation is proved immediately by taking the quadri lateral in the position A QDRs ; and now y = y = am ^K, so that squaring and adding leads to the desired relation. As in (ii.), quadrilaterals can be escribed to the circle BLB , which are inscribed in the circle OP, since N coincides with L. But the circles BNiB and BN*B are related to the circle OP with regard to poristic octagons; and the common tangents of these circles are easily recognised at the points PJ, p 4 , PJ. Conversely, starting with the circle cT and the internal point L, and drawing T^LT^ TJLT through L at right angles to each other, the tangents to the circle cT at T v T 2 , T 3 , T 4 will form a quadrilateral Q 1 Q 2 Q 8 Q 4 which is inscribed in a circle CQ, the diagonals Q^, Q 2 Q 4 passing through L, and being equally inclined to T^T 3 and jT 2 T 4 . If Q-fi, Q 2 c, Q^c, Q 4 c are produced to meet the circle CQ again in q v q z , q 3 , q, then q-^q 3 and g 2 g 4 are diameters of the circle CQ; for Q^ bisects the angle Q 2 QiQ 4 > so that the arc Q 2 g 1 = arc q-tQi, and similarly the arc Q 2 ^ 3 = arc g 3 Q 4 , so that the arc q l Q 2 q 3 = &YC yiQ$3> an( i eac ^ ^ s therefore a semi-circle. It follows, from elementary geometrical considerations, that LT* + LT* +LT 3 *+ LT? = 4r 2 , or T^ , so that cq* + cg 3 2 = cq* + cg 4 2 = (^ 2 - c 2 ) 2 /r 2 , leading to 2 (R 2 + c 2 ) = (^ 2 - c 2 ) 2 /r 2 , or, as before, +-J= 1. FOR ELLIPTIC FUNCTIONS. 127 t> Denoting by u v u 2 , u 3 , u^ the values of u at Q lt Q 2t Q 3 , Q 4 , so that u^ - u. 2 = u 2 - u 3 = u 3 - v, 4 = \ K ; and denoting by cZ lf c^ 2 , cZ 3 , cZ 4 the coiTesponding values of dn u, then ( 5 7) d 1 d 3 = d 2 d^ = K ; and ( 120) ZQ = 2Z(l-/ c / )dnu, so that " " ^ " while = 21(1 - K )(d^) ; Fig. 17. Now by a property of the circle (Euclid VI. D) so that - ^^(^i + d s )(d 2 + cZ 4 or (^i + ^ 3 )(^ 2 + cZ 4 ) is constant, and =2 N //c / (l+ the value obtained by putting u 4 = 0, when and d* = 128 THE ADDITION THEOREM Then when Thus so that 127. Poristic Pentagons, etc. (v.) When v = \K, or -fTf, the poristic polygons are pentagons (fig. 18), and the relation to be satisfied is of the form or p-q=p + q-i where p and q are used to denote r/(R c) and r/(E + c). We notice that the relation for pentagons leads to a cubic equation, when two of the three quantities R, r, c are given ; but the equation reduces to a quadratic when c = or the circles are concentric, the case considered by Euclid. The reader is referred to the articles of Cayley (Phil. Mag., Series IV., Vol. 7, and Collected Works) and to Halphen s Fonctions Elliptiques, t. II, chap. X., for the proof of this relation and the similar relations for other polygons. We shall find that Halphen s a and y (t. II, p. 375) are con nected with our R, T, c, /c, and w by the relations v> -r -- (R + c) 2 r 2 * \R + c and thence Halphen s x and y can be formed. By the use of Legendre s Table IX. for F(<p, K ) (F. E., t. II.) we are able to construct geometrically, to any required degree of accuracy, figures of circles related to each other for poristic polygons of any given number n of sides. Having selected an arbitrary modulus K or modular angle Ja, we look out the value of K, and then determine, by pro portional parts, the value of in degrees corresponding to an FOR ELLIPTIC FUNCTIONS. 129 will mark amplitude of K/n, 2K/n, . . . ; and these values of the position of the points Q v Q 2 , .... Thus, in drawing figs. 13, 14, 16, 17, we have selected /c = sin 60, when K= 2 1565; and in drawing fig. 16 for poristic triangles, we find, from Legendre s Table IX., am J#=c.m. of 3S49 , amf^T=c.ir,. of 685 . A Fig. 18. These angles enable us also to set out figs. 13 and 14, where the circles are drawn so related as to admit of poristic hexagons. In drawing figs. 15 and 17, Landen s point L is sufficient to complete the diagram ; also to double the number of sides of a polygon of an odd number of sides. In fig. 18, K has been taken as sin 75, as in figs. 1, 2, 3 ; and now K= 276806 ; and from Legendre s Table IX., amifiT=c.m. of 3018 , am|/i=c.m. of 7020 / , by means of which the figures can be drawn. Fig. 19 shows poristic heptagons, to the same modular angle of 75, laid out by means of the relations 0! = am 4^=0.001. of 22S , 3 = am4A r =c.m. of 5649 , 5 = amffiT=c.m. of 776 . G.E.F. 1 130 THE ADDITION THEOREM 128. The poristic relation between the quantities R, r, c has been obtained by placing the polygon in a symmetrical position; but another method is employed by Wolstenholme (Proceedings London Math. Society, vol. VIIL, p. 136 ; also by Halphen, F.E.. II., chap. X.), where the polygon on the circle OP is considered in its limiting form, when passing through one or both of the common points B and B . Thus with triangles, the tangent to the circle oR at B must meet the circle OP again at a point PI, the point of contact of a common tangent of the two circles P and R, the degenerate triangle being BPP. For quadrilaterals, the tangents to R at B, B must meet at A on the circle P, BACAB being the degenerate quadrilateral. For pentagons we obtain the degenerate form BP^Pg_P t PB, where BP^ is the tangent at B to oR, the circle through B, JV|, B , and PT. is the point of contact of a common tangent of the circles OP and oR (fig. 18). For hexagons (fig. 16) the limiting form is BP^P^BP.P^B, where BP^, P^B f are tangents at B, B to the circle through B, JVi, B r : and so on. FOR ELLIPTIC FUNCTIONS. 131 129. Geometrical Applications of Elliptic Functions to Spherical Trigonometry. Taking the fundamental formulas of Spherical Trigonometry cos c = cos a cos b + sin a sin b cos (7, sin A sin B sin C = . -= - = K , suppose; sin a sin 6 sm c then cos C = ^/(l /c 2 sin 2 c) = Ac, so that cos c = cos a cos b -f- sin a sin 6 Ac, a formula like that of 119, with a, b, c for 0, ^, y ; so that if, keeping (7, c, and therefore K constant, we vary a and b, then cos B . da + cos A .db = 0, or dafAa db/Ab = ; and, conversely, the integral of this differential relation is the formula above. (Lagrange, Theorie des fonctions, p. 85, 81, 82 ; Legendre, Fonctions elliptiques, t. I., p. 20.) If, in Jacobi s notation, we put a = am(u, K), b = am(r, /c), c = am(vj, /c), then the differential relation becomes du dv = 0, so that u v = & constant = w, since a = c, or ^t = 10, when & = and v = 0. Supposing K is less than unity, and the angle C is acute, then c>C, and of the other angles, one, A, must be obtuse, and the other, B, acute. But by changing to the colunar triangle on the side BC, we may convert the triangle ABC into one in which all three angles are obtuse ; and in such a triangle we may put a = am u, b = jr am v = am(2 J fiT v), c = am(2^T w) ; so that if the triangle ABC has three obtuse angles, we may put s , where it 1 -f- u. 2 + U B = u + 2K v + 2K lu = and now cos A = dn u v cos B = dn u. 2f cos C = dn u, so that, by 29, we may write A = TT amC/citp I/AC), B = -TT am(/at 2 , l//c), C=TT am(/cW 3 , l//c), where /c is less than unity. 132 THE ADDITION THEOREM For instance, if ABC is the spherical triangle formed by three summits of a regular tetrahedron, A = B = C = ITT, and cos a = cos b = cos c = J, sin a = sin b = sin c = f >^/2, sin a 4^2 8 * 8 * ** ~ 16 while ^ = ^2 = ^3 = j^K, so that en K= - }, sn |JJT= f */2, dn 4^= f . When /c = 0, K=\-w, and the triangle J.5(7 is coincident with a great circle ; and now When K = 1, K=vo; and therefore of u v u 2 , U 3 , two of them, say i&j and i& 2 , are infinite ; so that cos a = sech u x = 0, or a = JTT ; and similarly b = JTT ; the triangle ABG now has two quadrantal sides and therefore two right angles, the third side c and angle G being equal, and taken greater than a right angle. 130. For values of K which would be greater than unity, we change the notation by considering the polar triangle; and now if ABO is such a polar triangle, having three acute sides, instead of three obtuse angles, we put sin a _ sin b _ sin c _ sin A sin B sin C and A = am v lt B = am v 2 , 0= am V B , where v l = 2Ku l , v 2 = 2K u- 2 , v% = 2K u s , so that ^ + ^ + ^ = 2^. Now sin a = K sn v v sin b = K sn v 2 , sin c = K sn v 3 ; cos a = dn v v cos b = dn i> 2 , cos c= dn V 3 ; so that ct = am(/c^ 1 , l//c), & = am(/ci> 2 , l//c), c = am(/c^3, l//c). The fundamental formula cos c = cos a cos b + sin a sin b cos c now leads to the formula of 121, dn V B = dn ^dn v 2 +/c 2 sn ^sn u, en v s , or dn^ + v 2 ) = dn Vjdn v 2 /c 2 sn ^sn ^c^Vj + V 2 ). In the degenerate case of /c = 0, K=^-JT ) and and now a = 0, 6 = 0, c = 0, so that the spherical triangle is FOR ELLIPTIC FUNCTIONS. 133 indefinitely small, and may be considered a plane triangle; and we can thus deduce the formulas of Plane Trigonometry. 131. A spherical triangle thus falls into one of two Classes, I. or II. ; in Class I. the triangle, or a colunar triangle, has three obtuse angles; in Class II. the triangle, or a colunar triangle, has three acute sides ; the quadrantal triangle falling into Class I., and the right-angled triangle into Class II. In Class I. we put sin -4 _sin jB_sin C _ sin a ~ sin 6 sin c ~ and then K is less than unity; and we put a = am u l} b = am u.- c = am u s , where u l + u 2 + u 3 = K, and then A = TT Sim(KU v l//c), B = TT am(/cu 2 , l//c), C=7r am(/cU 3 , I/*). In Class II. we put sin a _ sin b _ sin c _ sin A ~ sin B ~ sin G~ and then K is less than unity ; and we put A = amv 1 , .Z? = amt 2 , 0=ami i s , wh ere v l + v 2 + v 3 = 2K, and then a = am(/cf 1 , l//c), 6 = am(/ci 2> l//c), c = am(/ci 3 , l//c). When this triangle of Class II. is the polar of the triangle in Class I , u x + 1\ = u. 2 + v. 2 = u 3 + v 3 = 2K. The change from one Class to the other affords an illustration of the change from one modulus to the reciprocal modulus ( 29). The spherical triangles employed originally by Lagrange and Legendre fall into Class I.; and a full discussion of the connexion between Elliptic Functions and Spherical Trigono metry will be found in the Quarterly Journal of Mathematics, vols. 17, 18, 19, in articles by Glaisher and Woolsey Johnson. But it is preferable in some respects to work with the spherical triangles of Class II., as growing out on the sphere more naturally from the infinitesimal plane triangle ; so it is proposed to develop here the relations with Elliptic Functions by means of a typical triaDgle of Class II., having three acute sides, and to refer to the articles of Glaisher and Woolsey Johnson for the corresponding relations of Class I. 134 THE ADDITION THEOREM 132, Writing c v s v cZ x for cnv l} snv v duv v etc. : then with v l +v 2 +v 3 = 2K > we may put, in Class II., A=a,mv l , B = &mVz, (7=amv 3 ; so that cos A = c v sin A = s v etc. ; and now sin a = K sin A = KS I} cos a = d v etc. From the fundamental formulas cos c = cos a cos b + sin a sin b cos C, cos 0= cos A cos 5 sin A sin 5 cos c, we obtain d = d where <i 3 = dn i; 3 = dn(^ 1 + v 2 ), c 3 = en v 3 = cn(v 1 + 1 2 ) . Again, from these two formulas of spherical trigonometry, cos (7= cos A cos B sin jl sin .B(cos a cos 6 -f- sin a sin 6 cos (7), o ^ _ cos A cos 5 sin A sin 5 cos a cos b - COS L/ --- - - ; - - - ; - - ; - - ; - - - - 1 sin A sm B sin a sin 6 so that -CD- . ., -, cos a cos 6 sin a sin 6 cos A cosB Similarly, cosc = = : ^ ; --- : 5 - -, 1 sm J. sin B sm a sin 6 leadin to d = dv 12 As a specimen of Class II., take the spherical triangle formed by three adjacent summits of a regular icosahedron ; then A=B=C=fr; , cos C+ cos A cos B cos (7 1 and cosc = : : ^ -- = T~ n n=> sin J. sm B 1 cos u ^75 so that K = sine/sin C = 1^7(10 2^/5); and then ^ = ^2 = ^3 = !^, so that en K =cosC= K^o - 1 ), dn f K = cos c = -J-^/5. 133. To prove that in a triangle of Class II. we obtain the differential relation cos b. dA + cos b. dB = 0, or dA/AA + dB/AB = 0, when we change A and 5, keeping c and (7 constant, dis place the triangle ABC into the consecutive position ABC\ keeping the points A, B fixed and the angle AC S unchanged in magnitude (fig. 20). FOR ELLIPTIC FUNCTIONS. 135 Then, if CA and CB produced on the sphere meet the great circle of which C is the pole in P and Q, the arc PQ = G ; and if C A and C B produced meet this great circle in P f and Q , the arc P Q is ultimately equal to the arc PQ, or Fig. 20. Fig. 21. But PAP = -dA, QBQ = dB; while ultimately PP = -sin 4P . cZJ. = -cos 6 . dA, QQ = cos a . dB; so that cos b . dA + cos a . dB = 0, or since sin a = KsinA, With A = am v v B = am v z , this becomes so that v x + v. 2 = constant 2K v^ where C = am v s ; since B+C=7r, or v. 2 + v% = 2K, when A = 0, i\ = 0. Conversely, this differential relation, interpreted with respect to the triangle ABC, of which the side AB is fixed, expresses the constancy of the opposite angle C. 134. If, as is customary, we deduce the differential relation cos B . da + cos A . db = 0, or da/Aa + db/Ab = 0, from a spherical triangle ABC of Class I., in which sin A=Ksiua, cosJ. = Aa, we keep the angle C fixed, and displace the side AB into its consecutive position A B , without change of length, through an infinitesimal angle 6 about the centre of instantaneous rotation /, the point of intersection of the arcs AI, BI, drawn perpendicular to CA, CB respectively (fig. 21). sin IBH cos B m , db ,, A A sm/J. Then --=11-=-. - da BB sin IB sin I AH cos A 136 THE ADDITION THEOREM 135. To obtain immediately the addition formulas (1), (2), (3) of 116 for the elliptic functions, Mr. Kummell draws the arc CD perpendicular to AB (fig. 20), and denotes the perpendi cular CD by p, the segments BCD, ACD of the angle C by F, G, and the segments BD, DA of the base C by / g ; so that F+G=C t f+g=c. (Kummell, Analyst, vol. V., 1878.) Now, from the right-angled spherical triangles ACD, BCD, cos G = sin A cos fr/cos p, sin G = cos A /cos p ; cos F= sin B cos a/cos p, sin F= cos J9/cos _p ; or with sin A = s v cos J. = c 1? sin a = ATS, cos a = d v etc., and writing M for cos p, cos G = s^dJM, sin G = cJM ; eosF=s 2 d l /M, siuF=c 2 /M. Also sin 2? = sin A sin 6 = sin a sin 5 = KS^, so that ^f 2 = cos 2 p = 1 - K \ V, a quantity which we have found it convenient to denote by D. Now, cos C= cos F cos 6r sin .F sin G, or c 3 - (s 1 s 2 cZ 1 cZ 2 - c^yD, or en (v + v 2 ) = en v 3 = (c t c 2 s 1 s 2 c? 1 cZ 2 )/D, formula (2). Again, sin Csii\(F-\- G) = sin jp 7 cos G 4- cos J^ 7 sin G, or s 3 = (s^gdg + sfrdJ/D, where 8 ft =8HV 8 =3Sn(^ 1 -|-V 2 ), as in formula (1). Changing the sign of v 2 , sn^-^) =sin( J F- G), or J^ 7 6r = am(v 1 v 9 ), while ^+(7 = am^ 3 = am(2^r t^ -y 2 ) = 7r-zm(v l + v 2 ), so that jP== JTT \ am( G = ATT - J am Thus, for instance, tan{ | am(^ 1 + v 2 ) + J am^ v 2 )} = cot (r = tan A cos 6 = s^d^jc^ tan { | am(i; 1 + v z ) J am(^ u>)} = cot J^= tan B cos a = s^djc^ Again, from the right-angled spherical triangles BCD, ACD, cos /= cos a/cos p = dJM, sin /= sin a cos B /cos p = KS^/M ; cos $ = cos 6/cos p = cZ 2 /Jf, sin g = sin 6 cos A /cos p = FOR ELLIPTIC FUNCTIONS. 137 and therefore dn(i\ + v. 2 ) = dn v s = cos c = cos(/+ g) = cos/cos g sin/sin g _d l d 2 as before, in (3), 116. Also sin(/+0) = /csn(v 1 + v 2 ), sin (/-</) = K sn(v l -v 2 }-, whence /and g can be found as functions of v^ + v^ and i\ v z . 136. The formula employed by Morgan Jenkins in the Messenger of Mathematics, vol. XVIL, p. 30, as fundamental in Spherical Trigonometry, is sm(A+B) _ sinO cos 6 + cos a 1 + cosc " and this now leads to ^1^9 ~r ^2^1 _ ^3 ~ or, in the Legendrian form sin G 1 + A(7 a formula already obtained from pendulum motion in 120. Then the formula S 1 C 2 ~~ S 2 C l S 3 d^-d^ l-cZ 3 or sin( J. B) _ sin G AB-AA 1-AO 5 gives sinU-J)_ sinO cos 6 cos a 1 cos c The formulas of 120, in the form 1 -f c 3 c 2 c : lead to the relations sin(q + fr) _ sine C" ..................... cos 5 + cos J. ~ 1 -cos C" sin(q b) sin c cos B cos J. "~ 1 + cos (7 and from these four formulas of Spherical Trigonometry Mr. Morgan Jenkins deduces the analogies of Xapier, Delambre, and Gauss. 138 THE ADDITION THEOREM 137. Write, as before, in 135, A = am u, B = am v. F= \-K \ am(u + ^) + J am(u, / y), 6r = JTT J am(M- -j- i>) J am(u -y). Then, since siu(F 4- ) + 8in(F- G) = 2 sin .Fcos G, therefore, writing c v s v d v for en u, sn u, dn u, and c 2 , s 2 , cZ 2 for en -y, sn v, dn v, and D for cos 2 ^ or 1 K\\ 2 , sn(u + f) + sn(u v) = 2s l c 2 d 2 /I) ) ............ , ...... (1) cos(.F- G) - cos(,P+ G) = 2 sin J^sin G, cn(u v)+ cn(u + v) = 2 c^cJD ; ..................... (2) cos(/ (/)+ cos(f+g) = 2 cos/cos gr, dn(u + -y) = 2 d^dJD ; ..................... (3) in(^- G) = 2 cos ^ sin G, sin(u v) = 2 s^dJD ; .\ ............ ..... (4) cos(,P- G) + cos(T+ G) - 2 cos 7^ cos G, cn(u v) cn(u + v) = 2 s^s^dJD ; ................. (5) cos(f-g) cos(f+g) = 2sinfsing, dn(u + v) = 2 K \c 1 s 2 c 2 ID , ............... (6) sin(F- ^-sin^-sin 2 ^, sn(u - v) = (c 2 2 - c^VD = (s^ - s 2 2 )/D. ..(7) Again, since 1 + sin(/+ <7)sin(/-gr) = cos 2 ^ + sin 2 /, and sin(/+^) = /csn(u + ?;), sin(/ #) = * sn(i6 -y), -7;) = (^ 2 2 + /cVc 2 2 )/Z); ............ (8) - G) = sin 2 ,P+ cos 2 G, -v) = (c 2 2 + s 1 2 ^ 2 2 )/D; .............. (9) 1 - coa(F + G)cos(F- (7) = sin 2 G + sin 2 ^, 1+ cn(u + v) cu(u-v) = (c 1 * + c*)/D ;..... .......... (10) 1+ cos(f+g) cos(/ r/) = cos 2 /cos 2 ^, (11) -/c 2 sn(u + v) sn(tt -v) = (^ 2 + Ac 2 s 2 V)/D ; .......... (12) - sin(.F+ )sin(^- G) = siri 2 G + cos 2 F, (13) ; ........ ..(14) cos(/ g) = siu 2 f+sm 2 g, : , ........ (15) FOR ELLIPTIC FUNCTIONS. 139 {1 sin(.F+ G)}{lsm(F- G)} = (sin ^cos ) 2 , (16) (17) ............. (18) ............. (19) 8(^4- G)}{lcos(F- G)} = (sin Fsin G) 2 , (20) {1+ cn(u + v)}{! cn(u-v)} = (8 l d z +8 z d. 1 )*/D; ............ (21) (1 cos(/+0)}{l cos(/-(/)} = (cos/cos) 2 , -v)} = (d l dJ*ID; ................ (22) dv(u-v)}=K 2 (s l c< i +s 2 c l ) z /D; ........... (23) sin(F + 6r)cos(jF G} = sin G cos G + sin F cos F, sn (u + v) cn(i6 v) = (s&d^ + s^d^/D ; ......... (24) - sin(F- G)cos(F+ G) = sin G cos G - sin .Pcos F, sn(w v) cn(i6 + v) = (s&dz s&dJ/D ; ...... , ..(25) sin(/+ g) cos(f-g) = sin/ cos /+ sin g cos g, sn(u + v) dn(u v) = (s^c, + 8 2 d z c^)/D ; ......... (26) siv(f-g) cos( f+g) = sin /cos/- sin g cos g, sn(u v) dn (u + v) = (s-^cl^ s^d^sJD ; ......... (27) -cos(.F+ G)cos(f-g) = {cos A cosB-sinAsiuBcos(f+g)}cos(f-g), cn(u + v) dn(u - v) = (c^d^l 2 - K\s^jD ; ....... (28) cos(F- G) cos (f+g) = cos(F-G){ cos a cos b + sin a sin b cos(F+ G)}, cn(u v) dn(u + v) = (c^c. 2 d^d 2 + K ^S^/D ; ...... (29) eon 20= 2 sin 3 eos 0, sin{am(u + v) + &m(u v) } = 2 s^d^/D ; ................... (30) ) am (u v)} =28 2 c z dJD ; .................. (31) cos{am(u + v) + am(u - v)} = (q 2 - sfd^/D ; ............ (32) - cos 2F= sii^F- cos 2 ^, cos{am(^ + r)-am(u-t )} = (c 2 2 -s 2 2 c? 1 2 )/j[); ............. (33) the thirty-three formulas of Jacobi, given in his Fundamenta Nova, 18, and reproduced in Cayley s Elliptic Functions. 140 THE AUDITION THEOREM Similarly any other formula in Spherical Trigonometry is converted into a form of the Addition Theorem of the Elliptic Functions, and conversely; by writing c v s-^ for cos A, sin A, and d v KS I for cos a, sin a, etc., with Thus the six four-part formulas, of which cot a sin c = cot A sin B+ cos c cos B is the type, obtained by eliminating cos b between (a) and (J3), lead to Sg^ = s 2 c i + S i c 2^3> with five other similar relations. By means of these and the preceding relations we can prove the following examples on the formulas of Elliptic Functions. EXAMPLES. 1. Prove that, if u-\-v + w+x = 0, /. N en u dn v dn u en v , en w dn x dn w en x _ ~ snu suv snw s (ii.) /c 2 /cV 2 sn USUVSUWSD.X + /c 2 cn u en v en w en a? dn u dn t> dn w dn a; = 0. 2. Prove that ,. x , N x 2/c 2 sn u en i> dn v (I.) nsftt tn-f-8n( < tt+tj)= :r~9 -- ;r~ ? -- > dn 2 ^ dn 2 i6 (ii.) 1 - K 2 sn 2 O + v)sn 2 (u, - v) = (1 - /c 2 sn%)(l - /c 2 sn 4 ?;)/D 2 : (iii.) /c 2 sn(u + v)sn(u v)su(u + ^)sn(u w) - K 2 sn 2 v sn 2 ^(;) ~ 2 sn 2 (u l-snu_cn 2 i( ... (u.) 4. Prove that and hence prove that the expression 1 K sn x sn y 1 + K sn z sn w 1 + K sn x sn y 1 K sn z sn w FOR ELLIPTIC FUNCTIONS. 141 remains unaltered when for x, y, z, w we substitute respectively 5. Prove that, if tanh A = K sn 2 a, tanh B = K sn 2 /3, tanh(4-5)=*sn Deduce Jacobi s relations, or + y)sn(/3-y) 1 - /c sn(y + a)sn(y-g) l-/ y)sn(/3-y) l+/csn(y+a)sn(y- a ) or = 1 ; 1 - K sn(t - x)sn(y - z) ~L - K $n(t - y)sn(z - x) l-KSu(t-z)sn(x-y} l+KSn(t-x)sn(y-z) 1 + K sn( - y)sn(z - x) 1 + K sn(t - z)sn(x - y) or =1 ; (Glaisher, Q. J. M., vol. XIX., p. 22.) 6. Prove that the tangents at the points on an ellipse of excentricity e whose excentric angles are <p = JTT am(u, e), i/r = JTT am(i;, e), will meet on a confocal ellipse when u v is constant, and on a confocal hyperbola when u + v is constant. Hence show that the general integral of d<p/J(l - e 2 sin 2 0) - cty/JQ - e%in 2 ^) - may be written and convert this into the form cos y = cos < cos \[r+ sin sin ^-^/(l e 2 sin 2 y), proving that tan^V 7. Prove that the straight line joining the points ccu(u-\-v) } csn(i6 + -u) and ccn(u v), csr\(u v), on a given circle of radius c, will touch an ellipse whose semi- axes are c sn(/f v), ccnv, when u is constant and v is variable ; and determine the envelope when u is variable and v is constant. CHAPTER V. THE ALGEBEAICAL FORM OF THE ADDITION THEOREM. 138. The first demonstration of the existence of an Addition Theorem for Elliptic Functions is due to Euler (Acta Petropolitana, 1761 ; Institutiones Calculi Integralis), who showed that the differential relation connecting X = ax 4 + 4>bx s + Qcx 2 + 4?dx + e, or (a, b, c, d, e)(x, I) 4 , the most general quartic function of a variable x, and Y the same function of another variable y, leads to an algebraical relation between x and y, X and Y. This algebraical relation is _*JX\ 2 = a (x + y) 2 + M(x + y) + C, x y / where C is the arbitrary constant of integration ; and this relation when rationalized leads to a symmetrical quadri- quadric function of x and y, of the form ( 148) ax 2 y 2 + 2/3xy(x + y) + y(x 2 + xy + y 2 ) + 2% + y) + e = 0, or (ax 2 + 2/3x + y)?/ 2 + 2(/3x 2 + 2y + % + y^ 2 + 2&c + e - 0, or (ay 2 + 2/fy + y)x 2 + 2(/fy 2 + 2yy + S}x + y if + 2Sy + e = 0. (Cayley, Elliptic Functions, chap. XIV.) With a = and b = 0, X and Y reduce to quadratic functions of x and y ; and then *J X ~^Y =gi consfcant x-y is the general integral o 142 ALGEBRAICAL FORM OF ADDITION THEOREM. 143 139. By writing (lx+m)l(l x f +m ) for x, which is called a linear substitution, this symmetrical quadri-quadric function becomes unsymmetrical, the five constants a, /3, y, S, 6 being thereby raised in number to nine ; and then dx/^/X becomes changed to (lm f I mjdx I^X , where X = (a, b, c, d, e)(lx + m, I x + m )*. The invariants g<> and g z of the quartic X have been defined in 75, and in 53 the discriminant A=# 2 3 27<7 3 2 , and the absolute invariant J=gf/A ; and now, if g 2 , g^ A , J denote the same invariants of X , we find g. 2 =(lm -l myg 2 , gj = (l m-lmjg A = (Zm - Z m) 12 A ; while the absolute invariants J and J are equal. Conversely, any unsyrnrnetrical quadri-quadric function whatever of x and y may be written G(x, y) = (ay* + 2/3 y + y> 2 + 2(/3if + Zy y + S")x + yif- + H y + e" L, M } N being quadratic functions of x, and P, Q, R being quadratic functions of y. Then by differentiation (Px + Q)dx + (Ly+iM)dy = ; and by solution of quadratic equations Ly + M= *J(M- - LN) = JX, suppose ; Px+Q = J(Q 2 -PR) = JY, suppose; and thus we are led to the differential relation where X and T are quartic functions of X, not necessarily of the same form, but having the same g. 2 and g 3 . A linear transformation, such as that given by can however always be found, which will transform where T is a quartic having the same coefficients as the quartic X ; in other words, the quartics X and Y have the same in variants ; so that we may, without loss of generality, consider X and Fas of the same form, and therefore drop the accents in the expression for G(x t y}. 144 THE ALGEBRAICAL FORM Now so that *- = axy + p( x y a form of the integral relation, in which the coefficients a, 6, c, d, e in X and Y are functions of a, /3, y, S, e, determined by e), the Hessian, with changed sign, of (a, /3, y, (5, e)(x, I) 4 ; and 140. Lagrange proves Euler s Addition Equation as follows: Put dx/dt^^/X, and therefore dy/dt= ^/Y] then suppose; so that putting x + y=p, x y = q, then dp dq == x_ Y dt dt = %apq(p 2 + q 2 ) + bq(3p* + g 2 ) + Gcpq + 4>dq ; whence dt 2 dp d?p 2 dqfdp\ 2 c/p ~ = Both sides of this equation are now integrable, so that or We notice here that, if C=4b 2 /a, X- x-y OF THE ADDITION THEOREM. 145 141. In the canonical form considered by Legendre, with oc = suu, dx/du = ^/(l x 2 . 1 /c 2 ^ 2 ), y = sn v, dy/dv = ^/( 1 y 2 . 1 /c 2 2/ 2 ), then X = 1 x 2 . 1 A 2 , Y= ly 2 .! K 2 y 2 . Therefore dxj^/X + dy/J Y= 0, leads to du+ dv =0, or u+ v = constant; which, in Clifford s notation, may be written sn ~ l x -\- sn ~ l y = con s tant. Euler s Addition Theorem of 138 now gives _ (en u dn u en v dn v) 2 /c 2 (sn 2 & sn 2 i?) 2 (sn u sn vf _ /dn u en v en u dn i>\ 2 _ fdn(u + v) cn(u + v)\ 2 V snu sn v / \ sri(u + v) J by J. J. Thomson s formula of 121. 142. But the Addition Theorem (1) for sn(u + v) of 116, sn u en v dn i> + sn v en u dn u 1 when translated into the inverse function notation, gives 22? This reduces, for K = 0, to the trigonometrical formula the integral of and for /c = l, to tanh ~ l x + tanh ~ l y = tanh - l , l+xy the integral of cfo/( 1 - as 2 ) + c^/(l - y 2 ) = 0. Similarly, equations (2) and (3) of 116 may be written We can now see why so little progress was made with the Theory of Elliptic Functions, so long as the Elliptic Integrals alone were studied, and also why Abel s idea of the inversion of the integral has revolutionised the subject. G.E.F. 146 THE ALGEBRAICAL FORM 143. A slight change of notation in the canonical integral (11) of 38, suggested by Kronecker (Berlin Sitz., July, 1886), introduces a further simplification, on writing x = then dx/du = n /-t \ -j-9 = /C-( 1 -- )(1 KX) du 2 K with p = K - l -\- K - ) and now u=fdxj / JX, o with X = x( 1 px + x*). Now = sn - + sn 144. In Weierstrass s notation, we take X = x*-g. 2 x-g^ so that, in the general expression of the quartic X, a = 0, 6 = 1, c = 0, d=-lg 2 , e=-g^ , and now Euler s form of the Addition Theorem becomes, with z for G the arbitrary constant, Now if x = $>u, y = $v, so that then we shall find ( 147) that 2 = >(u + 1>) ; so that or, in the inverse notation, Put i6+f = ty, so that since ( 51) #>w is an even function, and p w an odd function of w ; then, with therefore also, by symmetry, OF THE ADDITION THEOREM. 147 Thus - $v $w <@w <@n> pu pv or ( pv - <$w }<>u + ( yw - $>u }<p v + (pu pv )$> w = 0, or (p v $> iv)pu + (p lv pu^pv + ($> u p v)pw = 0, 1, pu, p u, or 1, pv, <@>v =0 ........................... (G) Weierstrass thus replaces the three elliptic functions sni<,, en u, dn u by a single function pu, and its derivative p u. 145. Take for example the integral of ex. 8, p. 65, fX ~ $dx, where X=(x-a) (ax 2 + 2bx + c), a cubic function of x, having a factor x a. This example shows that we may put Z* ... ac-6 2 ^" Wlth ^=o, ^=^ a) and then Now, if y and are the values of x corresponding to the values v and w of u, and if = 0, or "Z - Sefo +Y~ kly +Z~ *dz = 0, then the integral relation (G) of 144 connecting x, y, z becomes (y-*)Z+(*-*)r4+( a! -y)Z*=0 ............ . ..... (1) We notice that the integral relation does not require the knowledge of the factor x a of X \ so that, writing we have, on rationalizing the relation (1), or X7Z={Aays+Btoz+zx+xy)+C(x+y+z)+I)}* ...(2) (MacMahon, Comptes Rendus, 1882 ; Q. J. If., XIX., p. 158.) Then sothat ff _ equivalent to Allegret s result (Comptes Rendus, 66). 148 THE ALGEBRAICAL FORM 146. We shall find it convenient to replace the constant O in Euler s integral relation by 4c + 4s, and to consider s as the arbitrary constant, the meaning of which is to be interpreted ; and then x-y or s = - * where F(x, y) = ax 2 y 2 + 2bxy(x + y) + c(x 2 + 4txy + y 2 = (ax 2 + 2bx + c)y 2 + 2(bx 2 + 2cx + c% = (ay 2 + 2by + c)x 2 + 2(6^/ 2 + 2cy + d)x + cy 2 + 2dy + e, a symmetrical quadri-quadric function of x and y. Treating s as a function of the independent variables x and y t we shall find 1 df ,y 1 dX , v ~ ~ (ax* + 3bx 2 + 3cx + d)y + bx* + Sex 2 + 3cfcc + e 9 /v , -c, /T7 2 / Y_|_ M- -^.JJ suppose ; z * (x y) and similarly we shall find that \/Y has the same value. But if s is taken as constant, then or dxjJX + (%/</ F= 0, so that the differential relation which leads to Euler s integral relation is thus verified. 147. But now denote 4s 8 -02 s -0s b y 8 * where g% = ae 4<bd + 3c 2 , ^ 3 = ace + 26ccZ acZ 2 eb 2 c 3 , so that ( 75) g 2 and # 3 are the quadrivariant and cubicvariant of the quartic X (Burnside and Panton, Theory of Equations; Salmon, Higher Algebra). OF THE ADDITION THEOREM. 149 We shall find, after considerable algebraical reduction, that (x-yf so that 1 ^4- l dy - l ds - and the elliptic elements dx/^/X and dy/*J Fare now reduced by this substitution to Weierstrass s canonical form ds/^/S of 50. Mr. R. Russell points out a concise way of performing this algebraical reduction, by means of the linear substitution t = (TX + y)/( T +l) in the quartic (a, b, c, d, e)(t, I) 4 ; which then becomes of the form Xr^4 i (X l y + X^ + QF(x t y^+4 ! (Y 1 x + F 2 ) T + F, or ^T 4 + 4T 3 + 6CT 2 + 4DT+#, suppose. If the invariants of this new quartic are denoted by (? 2 , G , then G, = (x - yYg G B = (x- y) 6 G B ; and $ = 4s 3 <s (r (x-y) (x-y)* 148. Rationalizing the integral relation of 146, or s 2 (x - y) 2 - sF(x, y) - E(x, y] = 0, where E(z, y) = {(ac-b 2 ) + i(ae - c 2 )2/ 2 + (be - cd)y + ce - d* ; <* ( -Artfc-yF-rffa y)-H(x, y) = o where ^(ic, y) = (ac b 2 )x-y 2 + (acZ bc)xy(x a symmetrical quadri-quadric function of x and y. 149. When aj = y, F(x, x) = X, and ^(aj, a;) = H(x t x) = (ac - b^ + 2(ad - bc)x s + (ae + 2bd - + 2(be- the Hessian H of the quartic A r . 150 THE ALGEBRAICAL FORM One value of s is now infinite, and the other t~ as in 75 ; for, when x = y, F(x,y)-JXJY = 2(x-y) 2 _ 1t {F(x,y)}*-XY . -ZE(x, y) H %x-yY{F(x, y)+JXJY}- ll F(x, y) + JXJY~ ~ X a substitution due originally to Hermite (Crette, LII., 1856). Now, since t = GO , when X = 0, or x = a, (x-y)* Q ft-\ -H/X), a denoting a root of the quartic X = ; and here T=*/(*P-9J-9J (I>+ YX-X+XY =lt _ (x-y)*{(Y } x+ where G is a certain rational integral function of x of the sixth degree, called the sextic covariant of the quartic X ; the preceding algebra showing that T 2 Z 3 =G 2 , or 4# 3 -# 2 //Z 2 +# 3 ^ 3 + 2 = 0, ......... (H) this is called a syzygy between X, H, and G. (Burnside and Panton, Theory of Equations, p. 346.) For instance, if X is already in Weierstrass s canonical form, so that, if x = $u, X = #/ 2 u = 4a; 3 - g 2 x - g s , then H= and now t = so that ^ = This may also be written pStt^ftt- 150. Withy =00, 2s = ax 2 + Zbx + c - or s 2 - (ax 2 + 2bx + c)s- (ac - 6 2 > 2 - (ad - bc)x - J (ae - c 2 ) = 0. With y = Q, 2s = (ex 2 + 2dx + e- +Je^X)/x 2 , or cc 2 s 2 - (ex 2 + 2dx + e)s-(ae c 2 )x 2 - (be - cd)x -ce + d 2 = 0. OF THE ADDITION THEOREM. 151 Writing F(x, y) in the first equation of 146 in the form Y+ J Y (x - y) + T V Y"(x - 2/) 2 , we can find x as a function of s and y by the solution of a quadratic, in the form ..- This method of the reduction of the general elliptic element dx/^/X to Weierstrass s canonical form ds/^/S is taken from a tract " Problemata quoedam mechanica functionum ellipti- carum ope soluta. Dissertatio inauguralis" 1865, by G. G. A. Biermann, where the formulas are quoted as derived from Weierstrass s lectures. 151. Changing the sign of ^/F, we find that .*fe.y)WA/r 2(x-y)* leads to the differential relation 1 dx 1 dy _ 1 ds f ~~ = so 9 implying that u v = when x = y, since s oo when x = y : and now, in Weierstrass s notation, r that, putting /d( rx r*> u v = ldx\JX /ds/^/S, ^/ +/ Changing the sign of v, and therefore again of F, so that p2it = - H X /X, $>2v = - H y / F, implying that u = when X = 0, v = when F=0 ; so that where a denotes a root of the equation X = 0. Then w 152 THE ALGEBRAICAL FORM Mr. R. Russell finds, as is easily verified algebraically, that ___ (x-yy X" (x-y)*X (x-yf Y" = But, from the Addition Theorem (F) of 144, and therefore 2 p(uy)- the sign being determined by taking v small, when ^/ = a, nearly. Now, p / (u-i;) so that, as in 147, 152. p2v = - It H y l Y = (6 2 - oc)/a, and p 2v = - It G v / Y% = (a?d - 3abc + 26 8 )/a l ; ax + b Again, from equations (F)* and (G) of 144, _ Y&+ Y 2 ~ and putting u = 0, and therefore x = a, we find aa -\- b _ p v + p 2v ^/a <pv <p2v so that the quartic can be solved, when <@v and p v are known. (Solution of the Cubic and Quartic Equation, Proc. London Math. Soc., vol. XVIII., 1886.) OF THE ADDITION THEOREM. 153 Otherwise, with t= -H/X, d^ = _H X-HX _ __2 dx ~^~ X while T 3 = 4 3 - g z t -g s = G 2 /X 3 , so that d t/JT = - Zdx/JX, and u = /dx/^X = $/dt/JT= ^ ~ \ - H/X), a denoting a root of the quartic X=0. Then p2u = t = - H/X, & 2u = -T=- G/X? ; while v = when y = a, and Y= ; so that ptt = , = .^?) 2(x a) 2 u= /S= ( aaS + ^^ q2 + ^ Ca + ^ X + ^ 4 ^ Cq2 + IX (x-a) B ..** If v, k, K denote the values of u, s, S, when x = oc , ^ = J (aa 2 + 26 a + c) = p v, J^ = (aa 3 + 36a 2 + 3ca + cZ)^/a = - p v 7 aa 3 + 36 S K = x a so that *_= __ = and now p2v = (b--ac)/a, p 2v = (a-d Conversely, given these values of p2v and p 2-y, and supposing the bisection of the argument of the elliptic functions to be carried out, we can determine %>v and p i , and thence solve the quartic equation X = 0. 153. Since F(x, a) vanishes when x = a, a root of X = 0, it is divisible by x-a ; so that , suppose, x ~~ a a typical linear transformation, which converts dx/^/X into ds/^S, the canonical form of Weierstrass. Denoting the four roots of X = by a, /3, y, S, then since 6/a= -i( a we may write = jL f a- x-a a -/3 a-7 a-6V 154 THE ALGEBRAICAL FORM and now = /3(a - y)(a - 8) + y(a - 3)(a - ) + ffa - /3)(a - y) a with three other values ft, y , S corresponding to ft y, S. Now ^/S=- -j-^- TO (x-aY JX ^^ Denoting by e 1} e 2 , e 3 , the roots of the discriminating cubic 4<e*-g 2 e-g 3 = Q, so that $=4(s gjXs e 2 )(se 3 ), then we may write r _ Q s- ei = \a(a -y)(a-S ) ^, C OL JU OL so that, to x = a, /3, y, S, corresponds s GO, e v e 2 , e 3 ; and then - y) }, If we interchange a and /3, and put then to z = /3, y, (5, a, corresponds s 1 = oo , 6 3 , e 2 , ^ ; so that s = s l gives a linear substitution converting dx/^JX into dzjJZ, in which 05 = a, /3, y, S, corresponds to z = /3, y, ^, a. If s is replaced by pu, and the same function of z by p;, then we find from 54 that V = U, U + 0) v U + CO-L + ft) 3 , U + 2ft)! + fc> 3 , gives the four linear transformations which leave dx/^/X unaltered ; and corresponding to the values (a, /3, y, 8) of x we find (a, ft y, 5), (A y, 5, a), (y, A a, /3), (ft a, ft y) of ; the first transformation being merely z = x, not a distinct trans formation. OF THE ADDITION THEOREM. 155 154. When, as at first, _ and when e is a root of the discriminating cubic, then s e is a perfect square ; and we find where, as in 70, the quartic X is resolved into the quadratic factors N x and D X) and Y into the corresponding factors N y and D y ; this can be done in three ways, corresponding to the three roots of the discriminating cubic. Thus the integral relation x-y leads to the differential relation as is easily verified algebraically, N and D being quadratics. 155. A more elegant expression can be given to these rela tions if we follow Klein (Math. Ann., XIV., p. 112 ; Klein and Fricke, Elliptische Modulfunctionen, 1890) in employing homogeneous variables x and x. 2 , by writing xjx 2 for x, and 2/1/2/2 f r y > an d now /dx 7x~j Conversely, by writing x for x v and 1 for x 2 , we return to our original non-homogeneous variable x. Klein employs the abbreviations (xdx) for x z dx l x 1 dx z , and (xy) for x^ also fx for (a, 6, c, d, e)(x v # 2 ) 4 ; and now with 156 THE ALGEBRAICAL FORM and reducing to the above in 153, when fz/ = 0. The Hessian JT or ZT( 15 03 2 ) of X or f(a) 15 a; 2 ) is now given by and the sextic covariant G or G^, a5 2 ) by aff We may also use x and y as the homogeneous variables in the quantities, instead of x and x 2 . Thus, for example, the integral ff~&(xdy), where f = x ll y -f 1 lx Q y Q xy 11 (the icosahedron form) is shown to be elliptic by means of the substitution where dxdy Then we can verify the syzygy where T= - Now since g s =^T 2 ~ 5 , provided <? 3 = OF THE ADDITION THEOREM. 157 f l xcly 6 f x dz 6 6 j w=u^^^=^- iz -^ m ^ Similar reductions will show that the interals are also elliptic ; also the integrals f(tfy - xy 5 ) ~ \xdy) and depending on the octahedron form, (Schwarz, Werke, II., p. 252 ; Klein, Lectures on the Icosahedron.) "- ! 156. The further development introduces the theorems of Higher Algebra on the quartic and cubic, for the treatment of which the reader is referred to Salmon s Higher Algebra and Burnside and Panton s Theory of Equations. Thus, H denoting the Hessian of a quartic X, and e v e 2 , e 3 the roots of the discriminating cubic 4e 3 g 2 e g B = 0, then 4(H+e l X)(H+e,XXH+e.,X) = 4 ! H 3 -g,HX- 2 +g B X s = - G\ where G denotes the sextic co variant ( 149) ; so that H + eX is the square of a quadratic factor of G. Following Burnside and Panton (p. 345) we shall find it convenient to put 16(H-{-eX)= P 2 ; and then P 1 P 2 P 3 =32G, P p P. 7 , P 3 denoting the quadratic factors of the sextic co variant G. Then P, 2 + P 2 2 + P 3 2 = - SH, since e x -f e 2 + <? 3 = ; while (e, - e 3 )P^ + (e, - eJP,* + (e, - e 2 )P 3 2 = ; and eJP* + e 2 P 2 2 + e 3 P 3 2 = - 1 6(^ 2 + e* + ef)X = - 8g. 2 X. Since (e 2 - e 3 )P 1 2 = (^ - e 3 )P 2 2 - (^ - e 2 )P 3 2 therefore each of these factors must be the square of a linear factor, and we may therefore put so that Uj and u 2 are linear ; and now 158 THE ALGEBRAICAL FORM 157. Mr. R. Russell points out (Q. /. M. t XX, p. 183) that Hermite s substitution of t = H/X reduces the integral dt cu &ur , 3 _u dx~ "Jf 2 an ~~^ 2 ft 2? For so that G~^dx = J(4 3 g z t g 3 }~%dt. Again the integraiy*(4 3 g 2 t g B )~%dt, as well as the general (2) .(3) where K or K(x, y) denotes the Hessian of the cubic U(x, y), integral where 7 or U(x, 1) denotes the cubic (a, b, c, d](x, I) 3 , is again proved to be elliptic by the substitution given by , y} = dxdy "dotty* The cubicovariant J of the cubic U is given by ,(5) da? 1 dy and the discriminant A by A = a 2 ^ 2 + 4ac 3 -6a6ccZ+4c^ 3 -36 2 c 2 ; (6) and now we have the syzygy (Salmon, Higher Algebra, 192; Burnside and Panton, Theory of Equations, 159.) Differentiating (3) logarithmically 3ds_3JT_2T_ _&7 sdx~ K " U ~~ KU J while so that and dx U* sdx Uds -K = ~T ds (8) OF THE ADDITION THEOREM 159 When we know a factor, x a, of V, then we may employ, as in ex. 8, p. 65, the substitution s=U*/(x-a) ............................. (9) Putting U= (x - a)(ax 2 + 2V x + c ) then 4;2 3 g s is a perfect square, when ac -V 2 93 ~ and now z = 00 s + 26 a + c ~ aa 2 + 26a + c aa; 2 + 26 ar + c - # 3 (x - a) 2 -3K 3s (aa 2 + 26a -f c) U% ad 2 + 26a + c while 9/ 9(4s 3 + A) (aa 2 + 2ba + c) 3 ?7 2 ~ (aa 2 , ............. (10) a transformation equivalent to that of 47. 158. Mr. R. Russell also shows (Proc. L. M. S., XVIIL, p. 57), ,, where X denotes a quartic and H its Hessian, can be reduced to the sum of three elliptic integrals by Hermite s substitution t=-H/X. For we may replace ( 156) lx z + 2mx + n by 2xF\ + gP 2 + rP 3 or by 4p V( -H-ej:) + 4gV( ~ ^- ^0+ 4 ^ ( -JET- e 8 Z), where ^9, g, r are determined by equating coefficients ; while so that the integral becomes fp^-H-e^+q^-H-e^+rJt-H J aX + 3H . a X + H JXdt /3H 160 THE ALGEBRAICAL FORM the sum of three elliptic integrals. Particular cases may be constructed by making /3 and /3 zero, or a and a zero ; when we obtain f(lx 2 + 2mx + n)dx/X, or f(lx 2 + 2mx + n)dx/H. 159. Mr. Russell remarks that the reduction of the well- known hyperelliptic integral r (Ix^ + ^mx+ri^dx J x /( 1 X 2 . 1 + KX 2 . 1 + \X* . 1 K\X 2 ) to the sum of elliptic integrals is a particular case of this theorem, since the quartics 1 x 2 . 1 K\x 2 and 1 + KX 2 . 1 + \x 2 can be expressed in the forms aX + (3H and a X + fi H, by taking X= l+/cAa? 4 , and therefore H= K \x 2 ; and now a = l, a =l, fi= -(l+/cA)//cA, /3 X = (/c + A)//cA. These integrals are considered in Cayley s Elliptic Functions, chap. XVI., where x 2 is replaced by x; they arise in the expres sion of Legendre s elliptic integral jd<pl&((f), b) in the form E-\-iF, when the modulus b is complex, so that b 2 = e+if. (Jacobi, Werke, I., p. 380 ; Pringsheim, Math. Ann., IX., p. 475.) Writing P for x(I-x)(I+ K x)(l+\x)(l- K \x), Jacobi finds >, c)+F(<i>, 6)}, where or x 2( 5 A W ft - OF THE ADDITION THEOREM. Then employing the inverse function notation, fdx J P I ( ll+K.l+\.Xj\ . ll+K.l+X.X " /xdx _ ?~ When X is negative, then b and c are conjugate imaginaries ; so that we can now express F((/>, 6) in the form E+iF, when 6 2 is of the form e + if. For, writing - X for X, and now writing P for x(l-x)(l+ K x)(l-\x}(l -\-K\X\ f 2E f xdx - J JP *J(1+K.1-\) J JP th In the particular case considered by Legendre, X = 1, and now P = x(l-x^(I-A 2 ), on replacing K by /c 2 ; so that -x 2 . 1 - can be expressed by elliptic integrals. Mr. R. Russell employs the substitution and now dy f A(l-Bx)dx so that, putting x{(I+Bxy 2 -Ax}{(l+BxY--o-Ax}=P, therefore & = K -\ 2 , B= J( K \). Taking J B = x /(/cX), and ) 2 - Ax = (l-x)(l- K \x\ then 2J K \-A=-1- K \, and taking B= ^/(/cX), G.E.F. L 162 THE ALGEBRAICAL FORM 160. Mr. Roberts s integrals (Tract on the Addition of the Elliptic and Hyperelliptic Integrals, p. 53) where Q is a reciprocal quartic in x 2 , say or aQ = (ax* + %bx 2 + a) 2 - (2a 2 + W - furnish another particular case of Mr. Russell s theorem, since Q can be expressed in the form where X and H are in their canonical forms, y = dv Or we may put x-\-x l = u, x x~ l = v, when the integral becomes $A (U+V) + B( U- V\ where U= f du Thus where X = I+x\ H=x\ Therefore the integral / j -- ^- is reduced to elliptic integrals by a substitution, such as y = (I+x^/x 2 ; and then becomes Another particular case of the general theorem occurs in the reduction of the integral where R is a sextic function, the roots of which form an involu tion, and whose invariant E therefore vanishes (Salmon, Higher Algebra, 1866, p. 210). OF THE ADDITION THEOREM. 1G3 This invariant E is the one tabulated in the Appendix, p. 253, Higher Algebra, where it occupies thirteen pages. The sextic covariant G of a quartic X is a specimen of a sextic of which the roots form an involution ; and writing 32 G or c 2 ) (a 3 # 2 + 2b s x + 1 3 ) = a 1 .>. l .x 1 a 2 C "~ ^2 x ~~ ^-2) a s( x O 3 .x 3 ), then since the squares of P p P 2 , P 3 are linearly connected by the relation of 156, therefore P v P 9 , P 8 are mutually har monic, and any one is therefore the Jacobiari of the remaining two ; this leads to the three relations a. 2 c 3 -f 3 c 2 26 2 6 3 = 8 c 1 + c^Cg 26 3 6 1 = c^c., 4- a. 2 c t S/^i^., = 0. ^ 7 = are the six linear transformations which reduce to as in 74 ; so that if the quartic A" is resolved into the quadratic factors JV and D, we may write Now N/D is maximum or minimum when x = 9, or 0. Making P 1? P 2 , P 3 homogeneous by the introduction of y, which is afterwards replaced by unity, so that P=(a v b 1 ,c l )(x, y y- ..... then the three distinct linear transformations of 153, which leave dx/^/X unaltered, are found to be ^ _ _ KB J " ltyVx ~ty ~dx (R, Russell, Proc. L. M. S. t XVIII, p. 48.) Now i dx or f( Au i+ Su J( u 2 du i - *VK) y *JG J ^/{^^(V-V)} where u 1} u. 2 are defined in 155, is reduced by the substitution y 2 = uju v or p(x-<f>\ (x-0), 164 THE ALGEBRAICAL FORM This integral has been considered by Richelot (Crelle, XXXIL, p. 213) ; and by differentiation we find according as ?/ 2 is less or greater than ^/2-l ; and thence the integration can be inferred; the value of K to be taken is - 1 or tan 22 J, when it will be^found that K jK = 161. As further applications, consider the integrals where A0 = ^(1 - 6 2 sin 2 </>). (Legendre, Fonctions elliptiques, I., p. 178.) Putting A0 = ; 2 , and l-Z> 2 = c 2 , then 1 the integration required in the rectification of the Cassinian oval, given by or where r lt r 2 are the distances from the foci (a, 0). The expression 1 x*.x^ c 2 can be expressed by H 2 where X = x* + c, H=(l+c)x 2 ; and now the substitution y = X/H gives so that -16 V(2+2 C ) by means of the results of 39-41. In the Cassinian OF THE ADDITION THEOREM. 165 rZs 2 2 r 2 8 Now, if we put r* = (a 2 +/3 2 ) 2 cos 2 + (a 2 - /3 2 ) 2 sin 2 0, then s = a 2 /{ a 2 + /S^cos 2 ^ + (a 2 - /3 2 ) 2 sin 2 } Similarly which can be expressed in a similar manner. Again, substituting A 2 = # 3 , then particular cases of the preceding general integrals. Mr. R. A. Roberts (Proc. L. M. S. t XXII., p. 33) has shown that /(lx + m)(ax 6 + 2btf + c) ~ s or - j^ can be expressed as the sum of elliptic integrals, not always however in a real form. Mr. Russell shows that if x O v x 6. 2 are the factors of P v a quadratic factor of the sextic covariant, then lx+m is reduced by the substitution l f= p ( x -o i )/ (x ^e. 2 ) to the form lr-. R ^ * . , . dy, *//(a2/ 8 +26?/ 4 +c) " and this again by the substitution to the forn, two elliptic integrals, not necessarily however in a real form. 166 THE ALGEBRAICAL FORM Abel s Theorem, applied to the Addition Equation. 162. Euler s Addition Theorem is now found to be a very special case of a Theorem of great generality, due to Abel, the method of which we shall employ here, in the very limited form required for the Addition of the First Elliptic Integrals. Consider the points of intersection of the fixed quartic curve whose equation is 2/ 2 = A , ................................ (1) with any arbitrary algebraical curve whose equation in a rational form may be written f(,2/) = ................................ (2) By continually writing X for y* 2 , we can reduce equation (2) to the form P + Qy = 0; ............................ (3) and now the abscissas of the points of intersection of (1) and (2) are given by the equation P + QJX = 0, ............................. (4) or, in a rational form, P 2 Q 2 X = .............................. (5) Denoting the degree of this equation (5) by /*, and its roots by x v x 2 , ... Xfi, Abel puts ^x = P 2 -Q 2 X = C(x-xJ(x-x 2 )...(x-Xn), ......... (6) and now he supposes the roots of this equation to vary in consequence of arbitrary variations in the coefficients of the terms in equation (2), corresponding to arbitrary changes in the shape and position of this curve ; the coefficients in equation (1) are however kept unchanged. If dP, 3Q denote small changes in P and Q due to the changes in the coefficients, and if dx r denotes the correspond ing change in any root x r of equation (5), then Vx r . dx r + 2P<5P - 2Q8QX r = 0, or, making use of equation (4), = 0, dx r _ 9 QdP-P8Q_Ox L - ~ - suppose. Now, if the degrees of P and Q are denoted by p and q> then the degree of Ox is p + q , and we shall find this is always at least one less than JUL I, the degree of i//#, or two less than /m, the degree of \fsx. OF THE ADDITION THEOREM. 167 For if in equation (3), P 2 and Q 2 X are of equal degree, then q=p 2, and n = 2p , so that /x p q = 2; and pp q is greater than 2, if q is less than p 2. But if q is greater than p 2, then the order of \/ric is given by that of Q 2 X, and therefore /uL = 2q + 4<, while p = + l at most ; so that /u. p q = 3 at least. Since x6x is thus of lower degree than \fsx, we can split the fraction x6x/\fsx into a series of partial fractions, such that r x r 9x r m and now, if we make x 0, we find that a theorem in Algebra due to Euler ; otherwise stated as /y. W _ = 0- ..... (9) provided m is less than /z 1, the * marking the position of the missing factor x r x f . Applying this theorem to equation (7), we find so that, if, in consequence of any finite alteration of the coefficients in equation (2) or (3), the roots of equation (5) become changed to x\, x 2 , ..., x ^, then aX + ... +^dx^ = 0, ...... (11) the Theorem of Abel, as required for present purposes. It is the combination of the theory of Integrals and of the theory of Algebra which furnishes the key of Abel s Theorem ; the algebraical laws are expressed very concisely by a single equation (5), of which the variables are the roots, and whose coefficients are not independent, but are connected by a number of relations. Thus, if we take P of the ^ th order, and Q of the order p- 2, we have a plexus of ^ or 2p equations of the form (4) and the elimination of a, /3, /,..., y, ... leads to a determinant of 2p rows, each row of the form h--2 r /t- Jb r , . . . , JU r 168 THE ALGEBRAICAL FORM 163. Suppose for instance that (2) is the parabola (2) or (3) then equation (4) becomes ax z + 2px + y-JX = 0, ...................... (4) and (5) becomes the quartic equation (ax*+2px + 7 ) 2 -X = 0, ...................... (5) Denoting the roots by x v x 2 , x 3 , x^ then the elimination of a, /3, y leads to the determinant 4 > *^4 *i X/ 4 as the integral relation, corresponding to (/JL 4), & By making a = ^/a, so that the parabolas are of constant size, or by writing equation (5) in the form one root, x suppose, becomes infinite ; and now 4(/3 - % 3 + (4/3 2 + 2ay so that = 6c - 2y - or Now the two relations ^ + y - Vx/- Y i = 0, + y - JaJX t = 0, give by subtraction (x, - x z ){a(x 1 + x 2 ) + 2/3} = V S where (7 = 2aa; 3 2 + 46^ 3 + 6c 2 x /a /v /A r 3 ; and we thus obtain Euler s original integral relation, the general integral of the differential relation dx 1 /^/X l + dxJ^/X 2 = 0, when is constant ; and a particular integral of c^/V^i + dx 2 /JX 2 + dxJJXs - 0, when ic s is considered as variable. OF THE ADDITION THEOREM. 169 164. When X is in Legendre s canonical form 1 x 1 . l then Abel takes P = ax + x*, Q = b , and now equation (6) becomes - K 2 x-) where xf + x. 2 2 + x/ = b 2 K 2 - 2a, xfxf + x/x^ + a;^ 2 = b 2 + 6V + a 2 , r 2 r 2 r 2 _ 7,2 l X -2 X 3 - U . But a and b are determined by the equations ax l + ^j 3 + 6Z X = 0, aa3 2 + xf + 6.Y 2 = : so that b and therefore, as in formula (1), 116, > ^ 3 a^! JT , x. 2 X l 1 K-x^xf Also 1 - r/Y 2 . 1 - x, 2 " 1 - 3 2 - 1-6 V + 2tt + 6 2 + b 2 K - + a 2 - while ^ j 2 + a/V 2 + # 3 2 K*xfafa = 2a, so that 2 - a^ 2 - o; 2 2 - a; 3 2 + &x*xx = 2(1+ a) or (2 - x* - x* - xf + K*-x*-xx 3 *y = 4(1 - .r^Xl - 2 2 )(1 - ^ 2 ), which may also be written as in 119, with ^ 1 = snit, x. 2 = snv, x 3 = su(uv). This, with x^=siiu l , x 9 = snu 9t o: 3 = sn^ 3 , may be written 1 cn 2 i^ cn 2 i6 2 cn 2 ii s + *2 en i^ where u^ + it 2 ( 131); and, with a triangle of Class I., is equivalent to the formulas in Spherical Trigonometry 1 cos 2 a cos 2 6 cos 2 c + 2 cos a cos b cos c = /c 2 sin 2 a sin 2 6 sin 2 c = sin 2 ^. sin 2 6 sin 2 c = sin 2 a sin 2 jB sin 2 c = sin 2 a sin 2 6 sin 2 (7. 165. To obtain the Addition Theorem for Weierstrass s functions, we consider the intersections of the cubic curve y* = x*-g&-g s , or X, ............... (1) with an arbitrary straight line (2) 170 THE ALGEBRAICAL FORM Now, if x lt x 2 , x% denote the roots of the equation then so that a = v ^_ v ^ 2 , ft = ^ and ( 144) a; 1 +aj 2 +c 3 = The elimination of a and ft between these two equations and leads, as in 144, to the determinant (G) 1, x lt \/X l 1, pu, $ u 1, x 2 , ^/X% =0, or 1, pv, <@ v 1, x , ,JX 1, where u + v + iv = 0. In addition, from (5), so that 166. Consider the intersections of the fixed cubic curve with a variable straight line Then \}sx = (ax + /3) 3 ( Ax* + 3Bx 2 + 3Cx + D) and - - 3 /Y> ,7 /y> _ r^ a 3 -^ Denoting by y v y z , y. 3 the corresponding values of y, then { C+ J (a 8 - j + x 2 + a; as in 145. OF THE ADDITION THEOREM. 171 Now, if the constants a and /3 receive small increments Sa and S/3, then yf/x^dxi + 3(00?! + /3) 2 ( V + Sfl) = 0, and \/s x l = (a 3 A )(x l X 2 )(x 1 x s ), dx, =0, and the sum of the three integrals is a constant, which can be made to vanish by taking for the lower limits a root of the equation y = 0. In the particular case of the cubic curve the relation expressing the collinearity of the three points is aj 1 a? 2 ar s + 2/i2/ 2 2/3 =1 - Now, as in 145, with # 2 = 0, # 3 =1, and and, by symmetry, with (i we find from (F) 144, after reduction, so that it + ! = , a constant. With pa = l, then ( 149) p2a = l ; so that ( 62) p2a = ?(2a> 2 ~ a )> or a = ^2- We may therefore put U = Jct> 2 + , V = Jft) 2 #, and express x and ^/ by functions of t. For any other arbitrary value of a, the integral relation connecting x and y will be, by 145, and treatin as constant, this leads to the differential relation = 0. 172 THE ALGEBRAICAL FORM We can put (1-g*)* _(l -~ p * ~ where and pi0 = l, for the value z= oo ; and then 167. When the quartic JT is resolved into two quadratic factors N and D, we may replace (1) by the quartic curve y*=N/D; ................... ........... (i) and now equation (4) is replaced by PJD + QJN=0; ......................... (4) so that equation (5) becomes P*D-Q Z N=0 ........................... (5) The elimination of the constants from the plexus of equations determined by the roots of this last equation (4) leads to determinants, whose rows are of the form For instance, by taking P and Q linear, so that the variable curve (2) or (3) in 162 is a hyperbola, we can obtain the integral relation of 154 in the form - eontant . (W. Burnside, Messenger of Mathematics.) We have taken X as a quartic function of x, so as to apply to the elliptic functions, but Abel s theorem holds for any higher degree of X, the method of proof being exactly the same; and, according to Klein, we resolve X, supposed of even degree, into factors N and D, differing in degree by or a multiple of 4, when we wish to make use of the fixed curve 168. The reader is referred to the treatises of Salmon or of Burnside and Panton for the proof of the Theorems in Higher Algebra quoted here ; they are easily verified, however, if we work with the quartic in its canonical form U= x* 6m x 2 y 2 + y 4 ; when H = mx* + (1 3m 2 )x 2 y 2 my*, G = J(l - 9m 2 )xy(x* - y*). OF THE ADDITION THEOREM. The following examples, taken from recent examination papers, will illustrate the character of the algebraical work. EXAMPLES. 1. Denoting by U the binary quartic, reduced to its canonical form, # 4 6ra# 2 i/ 2 -f y 4 , its quadrin variant and cubinvariant by g. 2 and (/ 3 , and its Hessian and sextic covariant by H and G, prove that (i.) 4m 3 -# 2 m-(7 3 = 0; (ii.) H-t-mU is a perfect square ; (iii.) O Tf (iv.) H, ~ (vii.) the Hessian of XU+imH is (X 2 - T V(7X)ff + (i and the sextic covariant is 2. Denoting the roots of 4e 3 g 2 eg 2 = Q by e v e. 2 , e s , prove that the roots of (x- + %g. 2 } 2 2g 3 x = are of the form 3. Denoting the discriminant, Hessian, and cubico variant of a cubic / by A, K, and /, prove that (Work with the canonical form U=ax 3 -\-by 3 .) Denoting the same functions of XJJ-h/xG by A , K , J , prove that A = (X 2 -/A 2 A) 2 A, 4. Prove that X and Y in 139 have the same invariants and g s (Burnside and Panton, 1886, p. 418). 5. Prove that, in 156, Vfe - 8 >Pl + Vfe - *l) is the square of a linear factor of X. 174 ALGEBRAICAL FORM OF ADDITION THEOREM. 6. Discuss the properties of the quartic X in 153, whose roots are a , ft , y , 3 . 7. Prove that ( 160) O lt </>! ; 2 , 3 ; $ 3 , 2 > define an involu tion of the roots of the sextic covariant G (R. Russell). 8. Prove that the cubic substitution y = (bx s + 3e$ 2 + 3fZo; + e)/(ax s dy 3dx makes j-, ^7-^ TT~\ = ^ (fjMyg^Uy) where U x =(a, b, c, d, e)(x, I) 3 . (Hermite ; Crelle, LX., p. 304 ; R. Russell, Proc. L. M. S. t XVIIL, p. 52.) dx 9. Integrate /^4r 10. Prove that, with s = pu, u - e) - - (s 2 - 268 - 2e 2 - e 2 U = 8 -f 1 ~ ^ 1 *L_- ? s e 2 11. Prove that, if (i.) p(v; -20, -40) = 5, then p2v = 0, ^3v= (ii.) p(v; -60, -10) = 5, ............ 0, ......... i, (iii.) KO; -15, 19) =f, ............ I, ......... W, 12. Prove that (i.) /(A + Bx)dx/y is elliptic, if y 2 = (1 - x 2 )(a + 3a? (ii.) f(A+Bx+Cy)dx/lj- is elliptic, if t/ f(a;, y) = (a, 6, c, /, gr, ^)(^ 2 , 2/ 2 , 1). (W. Burnside). CHAPTER VI. THE ELLIPTIC INTEGRALS OF THE SECOND AND THIRD KIND. 169. The Elliptic Integrals, and thence the Elliptic Func tions, derive their name Elliptic from the early attempts of mathematicians at the rectification of the Ellipse. It was some time before mathematicians perceived that the simple integral to begin considering is which has not originally such a special connexion with the ellipse ; but the name Elliptic Integral has nevertheless been retained generally for all integrals of this nature. To a certain extent this is a disadvantage ; not only because we employ the name hyperbolic function to denote coshi(, sinl^t, tanh u, ..., by analogy with which the elliptic functions would be merely the circular functions cos<, sin <p, tan 0, ...; but also because it is found that the elliptic functions are a particular case of a large class, called hyperelliptic functions, but included in a larger class, called Abelian functions after Abel, which, beginning with the algebraical, circular, hyper bolic, and elliptic functions of a single argument u (jj = l) are in the general case the functions of p arguments which are met with when we consider the integrals /(!,, a* ..... x"- 1 ) dx/JX, arising in the linear transformations of J"dxj^/X, in which X is a rational integral function of x of the degree 2p + 2: for now the linear transformation (lx+m)/(l x+m f ) converts into (lm -V 175 176 THE ELLIPTIC INTEGRALS 170. Legendre s elliptic integral of the second kind has already been defined in 77 ; and denoting it by E$, then the length of the arc BP of an ellipse is given by aEtf>, where the arc BP and the excentric angle of the point P are both measured from the minor axes OB, and now the modulus is the excentricity of the ellipse. The quadrant of the ellipse BA is given by aE, where, r^Tr as in 77, E denotes /A0c?0, the complete elliptic integral of o the second kind, in which < = JTJ-. The perimeter of the ellipse is therefore ^aE, the same as that of a circle of radius aEfyir. The periodicity of sin < and A< shows that, as in 14, and generally when m is an integer. Expanded in ascending powers of the modulus 2 - so that, employing Wallis s theorems of integration, as in 11, T^ fj* i fi n ^?/1.3.5...27i-lV K- H H - E=J^ = ^\l-^^-^~^) -_J, o n ~ l whence the numerical value of E can be calculated. Tables of the numerical values of E(j> for every degree of < and of the modular angle are given in Legendre s F.E., II., Table IX. ; while the values of log E are given in his Table I. for every tenth of a degree in the modular angle. We reproduce this Table of logE, and of log^ 7 , correspond ing to the complementary modulus AC , to 7 decimals, and to every half degree in the modular angle J, corresponding to the values of logK in Table I, p. 10. 171. By differentiation and integration, we prove that d(E<j>\ F<j> d f .. fd<t> E<j> K 2 sin cos 0. zk?) ? d-^ F * }= J iv"^-?*- AT and therefore, with $ = JTT, dE K d E OF THE SECOND AND THIRD KIND. 177 o p o p >p p o p p 9 p p o Lt o >p p ip p ip o o p o p o o o o o p ip p ip o o p o p o 9 H 9 ! ip 9 o 9 ip 9 p 9 p 9 p 9 ip 9 ip 9 p 9 p 9 ip 9 ip 9 ip o p o p 9 >p 1 4n Th cc ci A, ^^-iHobbcixoc^t^iioui^^rcrc^c^^bbb N i>i>-i>i>-i>- i>-i>t t-i>.occooo^oooco;:;ooocooooccoo w H^ f illlliill||lilll|liis|g|gfii; i cot^ ~ o~ ^T^ rc?Ji p-Ho ~ < SSt>^S>O < ^cOTO<M-^oSSt^cD Icoco co QCGO oooo ccoo r^r^r^r^r^t^r^r^t--r^ p 9 p 9 o 9 p 9 o 9 ip 9 >p 9 ip 9 p 9 p 9 p 9 p 9 o 9 p 9 p 9 o 9 ip 9 ip 9 p 9 p 9 o 9 ip 9 p 9 !p 9 ip 9 ip 9 O CT5 CS CC 3C t^* r^* tO sO O O "^ "^ CC CC C^l ^ ^^ ^ O O C^ C^ CO CO t>* t^* SO CO 1^5 Li Ciooa5ooxxxxQccoxcoooooGocoxoooooooor^i^t^i> r^t^t> j> t^t^ o o o b o o o o o o 099909999999 tc G.E.F. 178 THE ELLIPTIC INTEGRALS We can now prove Legendre s relation, that EK +E K-KK is constant, and = JTT ; for denoting it by A, we find that dA/dic = Q, so that A is independent of K ; and taking K = 0, then o o 172. In Jacobi s notation, with < = E(f> = E am u =fdu. 2 udu ; o and now, from the quasi-periodicity of am u ( 14), where m is an integer. We may therefore, as in 78. separate E&mu into two parts, one the secular part, increasing uniformly with u, at a rate 2E per increase 2K of u, and the other a periodic part, denoted by Zi& in Jacobi s notation, and called the Zeta function ; so that or Zu =/(dn% - EjK)du. Addition Theorem for the Second Elliptic Integral. 173. A well-known theorem, due to Graves and Chasles, asserts that if an endless thread, placed round a fixed ellipse, is kept stretched by a pencil, the pencil will trace out a confocal ellipse (fig. 22). (Salmon, Conic Sections, 399.) If the excentric angles (measured from the minor axis of the ellipse) of the points of contact P, Q of the straight parts of the thread PR, RQ are denoted by <j>, \[s, so that the arc BP = aE(f), arc BQ = aE\js ; and if we put <p = am u, \}s = am v, the modulus K being the excentricity of the ellipse, then, as asserted in ex. 6, at the end of Chap. IV., R moves on a confocal ellipse, when u v is constant, and conversely. For the coordinates of R being given by cos \!s cos (h 7 sind> sinx^r x = a r-h- -- r-rS y = b .^ r\ sm(0 \Is) sm(0 \/s) we find from Jacobi s formulas (4), (5), and (31), 137, replacing u and v by ^(u + v) and ^(u v), OF THE SECOND AND THIRD KIND. cni> i sin (am u ami;) sn u sn v __ _ " " sn 179 -t;) am-?;) Therefore Fig. 22. , cn a = a dc ^ ~ where so that a 2 -/5 2 = a 2 -6 2 , and therefore E describes a confocal ellipse, if u v is constant. If 16 + v is constant, -we find (x/aj-(y//3r = l, where </ = a K sn |(u + v), $ = a K en l(u + v), so that a /2 + /3 2 = a V - a 2 - 6^ and ^ therefore describes a confocal hyperbola (MacCullagh). To realise mechanically this motion of R on the hyperbola, the threads RP, RQ must pass round the ellipse, and be led, in the same direction, round a reel rnoveable about a fixed axis at C ; so that, as the reel revolves, equal lengths of thread are wound up or unwound. If the hyperbola starts from the ellipse at L, then 180 THE ELLIPTIC INTEGRALS If the threads are wound in opposite directions on the reel, then R will describe a confocal ellipse, as at first ; but in this case the reel may be suppressed, and the thread merely made to slide round the ellipse, as in the theorems of Graves and Chasles. Moreover, it is not necessary that the tangents RP, RQ should proceed to the same ellipse, but to any two fixed con- focals, and the same theorems hold. If tangents R P t RQ , are drawn to the ellipse from any other point R on the confocal hyperbola RR, forming with RP, RQ the quadrilateral RrRY, then r, r lie on a confocal ellipse, by the preceding theorems ; and now a circle can be inscribed in this quadrilateral whose centre is at T, the point of concourse of the tangents to the confocals at R, T, R, r \ for TR, Tr, TR, Tr bisect the angles of the quadrilateral ; (Salmon, Conic Sections, 189). If R is brought up to L, the circle touches the ellipse at L ; so that the point of contact of the circle inscribed in the area bounded by two tangents and the ellipse is at the point where the confocal hyperbola through the point of intersection of the tangents cuts the ellipse. 174. Putting u v = iv, or F<f> F\fr = Fy, then when v = Q and Q is at B, u = iu and P is at G where (j) = y suppose ; while R will come to 7) on the ellipse RD, where it is cut by the tangent at B. Now, since PR + RQ-arc PQ = BD + DG-&rc BG, or &rcPQ-wcBG = PR + RQ-BD-DG , therefore E$ E\}s Ey = a certain trigonometrical func tion of <p, \/s, y, which is found to be /c 2 sin <f> sin \fs sin y ; this is the Addition Theorem for the Second Elliptic Integral. -r? r>T>? ?f cosx^-cos^ 2 79 fsin0-sini//" /I For PR 2 = a 2 ! sin - ^ -- rf- J- + 6 2 ^ r-*r- -- TV~ COS r \ sm^-^) j \ Bin(0-V0 so that PR = aA0 - . while BD = r -- -. Sin y OF THE SECOND AND THIRD KIND. Therefore, by 121, sin y = a {ooa cos \lr -f sin sin i/^Ay cos(0 \f/)} l-A 2 y . = a - - L sin sm \/r sin y = a/c 2 sin sin \fs sin y. In Jacobi s notation this is written E am uE&rnvE am(i& r), or Zu Zr Z(u v) = K 2 sn -it sn v sn(u v). 175. Putting -y = w, and therefore i = Zw t then # am 2w 2E am w = /c 2 sn 2w sn 2/ w;, or changing w into Jie?, .Z? am iv 2E am Jtt; = /c 2 sn la sn 2 Jw; = sn w- ( 123). Then PjR + jRQ-arc P = BJD + DG-wc BG , , x l cni/j r, = a(l + an w) -- aE am i(; sn w sn w 1 dn \-asuw / sn w j-, f \ o /sn w dn to \ = 2a( ^am4te;)=2a( ^ ^amivj : \1 + en i/j / \ en -kiv / and now en \w t or en ^(u v) = b//3, where fi=OK. 176. A ready way of proving the Addition Theorem is to take the spherical triangle of Class II. , in which A = am v v B = am v 9 , C = am V B , wh ere i\ + v. 2 + 1 3 = 2/i , and to vary all the sides and angles, keeping K constant. Then dv 1 + dv z + dv s = 0, or dA /cos a + dB/cos b + cZ(7/cos c = 0, or cos 6 cos c . dA + cos c cos a . dB + cos a cos 6 . dC = 0, or (cos a sin 6 sin c cos J.)cLl + (cos 6 sin c sin a cos . + (cos c sin a sin 6 cos C)dC = 0, or cos ad A -f cos 6c?5+cos ccZO os J.cZ^ = K 2 d(siu A sin B sin (7). 182 THE ELLIPTIC INTEGRALS Integrating, E(A ) + E(B) + E(C) - 2E= /c 2 sin A sin B sin 0, since y cos adA =/J(i - K *n*A)dA = E(A), o and v 2 = makes B = 0, and ^4 + (7= TT, or A"( In Jacobi s notation E am ^ + E am t 2 + ^ am v s 2E = /c 2 sn ^s or Zv 1 + Zv 2 -f Zi> 3 = /c 2 sn with ^ + v 2 + v 3 = 2if. With or Zu + Zv Z(i6 + v) = /c 2 sn u sn v sn (u + v). Fagnano s Theorems. 177. The particular case of the -Addition Theorem, obtained by putting y = j7r, or u v = K, was discovered by Fagnano (1716), and leads to his theorems, namely, that if P, Q are two points on an ellipse of excentricity K, whose excentric angles 0, \fs, measured from the minor axis, are such that A0 AI/A = K, or tan tan \{s = l//c = a/b, then the arc BP + arc BQ arc J.5 = /c 2 sin sin i/^, or arc BP arc J. Q = a /c 2 sin sin i/r = Ax /a ; x i x /: 2. a 2 and then tan 2 tan V = ^^-__- 2 ^ = _, or K 2 x V 2 - ct 2 ^ 2 + ic /2 + a 4 - 0. On reference to tig. 23 it will be found that, if OF, OZ are the perpendiculars on the tangents at P and Q, then (i.) ^40^=0, AOY=^, (ii.) wcBP-sircAQ = PY=QZ=VQ-PT, - so that VZ=PT, and PFor QZ=K*xx /a , the tangents at P, Q meeting 0-4, 01? in T, F; (iii.) OP 2 -OQ 2 = OF 2 -0 2 ; (iv.) OY.OZ=ab. When P and Q coincide in F, then P is called Fagnano s point ; and then (i.) the arc BF arc AF= a-~b\ Va? - T-, (iii.) OF THE SECOND AND THIRD KIND. 183 (iv.) the tangents at P, Q intersect in R on the confocal hyperbola FED, through F, D, whose equation is (v.) the tangents at P and Q intersect in R on the confocal ellipse KDH, through K, D, H, whose equation is (vi.) PR-eL (vii.) the circle inscribed in the region bounded by A D, DB and the ellipse AB touches the ellipse at F; etc. The proof of these theorems is left as an exercise. o Fig. 23. 178. Denoting the arc J.Pby s, the perpendicular OFon the tangent at P by p, the angle A T by \fr, then by Legendre s formula ds d? so that s + PY= /pd\[f ; and in the ellipse p = while P F= dp[d\fr = a/c 2 sin ^ cos i/r/ A\^ = a/c 2 sin sin \f, ; so that s + a/c 2 sin ^ sin ^ = afk\}sd\{s = aE\fs = arc 5Q, or arc J2Q arc AP = a/c 2 sin ^> sin i/r, as at first, in Fagnano s Theorem. 184 THE ELLIPTIC INTEGRALS Confocal Ellipses and Hyperbolas. 179. If we put then x = c sin cosh 0, y = ccos(J> sinh 9 ; T> 2 n so that --^ + -0 =C 2 2 2 c 2 ^ > the equations of a system of confocal ellipses and hyperbolas, since cosh 2 $ sinh 2 6) = sin 2 + cos 2 ^ = 1. T , n dx 2 dy 2 dx 2 dy 2 2/ , 2 . 0jN 5^ + 4 5= d0 2+ ^ = c(cosh e ~ sin *> ; so that, in an ellipse BP, along which is constant, the arc BP = c/ / J(cosWO-sm 2 <j))d<p = aE</> as before, with a = c cosh 0, and the modulus equal to the excentricity sech 6. For the confocal hyperbola, along which < is constant, the arc is given by which can be expressed by elliptic integrals of the first and second kind, of Legendre s form. Putting the equation of the hyperbola is and now the coordinates of any point P on the hyperbola may be given by a cosec % b cot ^ ; and the tangent at P by and then amh 6 = JTT x, cosh 6 = cosec x, sinh = cot x, tanh = cos x, etc. The tangents at P, and at another point Q defined by will therefore meet at a point R, where a cosec x cot x cosec x cot x cos x cos x b cos x When we put v = am u, v = am v s\. * /\. the modular angle being 0, then as in 173 for the ellipse, OF THE SECOND AND THIRD KIND. SaCoCZ, Co Cn ifu - 1 1 ) 185 sn v) dn J(it v) _ b s 1 cZ 1 s 2 (i 2 SjcZg sn J (it + v ) d n J (it v) and therefore, eliminating en ^(u v) and dn i(u v), a en /csn /csn Ksn(tt+f) where a = and so that J describes a confocal ellipse, when u + v is constant. Fig. 24. 180. By putting u-\-v = K, we obtain theorems for the hyper bola (fig. 24) analogous to Fagnano s theorems for the ellipse. Now ( 123) a or a 2 and j describes the ellipse -FT), whose equation is o o x which will intersect the hyperbola in a point F, the analogue of Fagnano s point on the ellipse, the coordinates of which are c sin 0^/(l + cos 0), c(cos )*. 186 THE ELLIPTIC INTEGRALS Now, as in 57, with X = am U, x = am v, and and cot x cot % = K = cos 0, or sinhflsinh 6 = K, and if a;, y and a? , ^ denote the coordinates of P and Q, a; = a cosec x = a Ax7 cos x > ^ = a cosec x = ^Ax/cos x ; y = a cotx=a* tnx , 2/ = a cot x = a* tan x ; and thus yy = aY = C 2 cos 3 <. Drawing the perpendiculars Y, OZ from on the tangents at P, Q, and denoting the angles AOY, AOZ by o>, ; then cfcc w/6 2 tan co = .p = ~r-2 = tan </> cos x = tan < tanh = sin sin x /Ax ; sin ft> = sin < sin X , cosft> = Ax / , sin w = sin sin x, cos = Ax- Now denoting OF, OZ by p, p , then _p = ^/(a^cos 2 ^ 6 2 sin 2 ft)) = c^sin 2 ^ sin 2 o>) = c sin <p cos x ; pp = C 2 sin 2 0cosxcosx = C 2 sin 2 ^c Making use of the formulas ds d 2 p dp -j-= T^> p, and PF= - 7 , c^oj dw 2 l da) then PF- arc AP = also P F= c sin co cos (o/^sin 2 ^ sin 2 co) = c tan x Ax = c/tan x AX = c cosh sinh 0/^/(cosh 2 9 - sin 2 ^). 181. The arc AP of the hyperbola is now expressed in terms of an elliptic integral of the first and of the second kind ; we can however express the arc by means of two elliptic integrals of the second kind, or by two elliptic arcs by means of Lan- den s transformation ( 67). We shall find that if we put 2\ or sin2ir sin2\/r , (1-f sin0)A(i/r, y) then tan y = -r Y . . , sec v = / A sin , 4sin0 , where y 2 = , 7 = OF THE SECOND AND THIRD KIND. 187 _ n x ~ 0)A(Vr,y) ~ ( A , 1 + sin </> cos 2 , so that l + sin ^-sinX) ~ A x -. (l + sin0)A(^, y) cos a) + ^/(sin 2 ^ sinV) = AX + K cos x = (1 + sin 0)A(i/r, y) ; Integrating, (i and now the arc of the hyperbola 182. If we put then we find ( 180) = = 1 cos tan 2 ^ 1 cos sn = A / ^ > x __ l-(l-cos0)sin 2 x / _ A 2 x r +cos0 - and _ Now, sin(2 x - f ) = X sin g as in Landen s second transformation ( 123); and (1 + cos 0)A( X)c/ - ( A 2 + cos = 2A x d x + 2 cos , - siu Integrating, (1 +cos 0)^(f , X) = 2^x / + 2 cos <pF x - sin 2 ^ sin x cos xV^x j and the arc J.P can be expressed by means of E^ and E(, X). When x = x / = am i^ then ^^i^r; also ( 175) 2Y = ^(/c) + 1 - cos 0, while 2^ = -2" ; so that (1 + K )^(X) = ^(/c) + K K. 188 THE ELLIPTIC INTEGRALS 183. The following theorems, analogous to those of 177, can easily be proved by the student : (i.) The difference between the infinite asymptote DT and the infinite arc FT is equal to AD arc AF\ so that the difference between the infinite asymptote OT and the infinite arc AT is equal to OD + AD -2 &rcAF , (ii.) the coordinates of F are (c + &),/{(c-&)/c}, *J(V/c); and the tangent FK=AD = b, KG = c; (iii.) the tangents at P, Q intersect in R on the confocal ellipse through F, whose equation is ~ and the tangents at P , Q intersect in R on the con- focal hyperbola through D and K, whose equation is = c; c a a (iv.) PR - arc PF=QR- arc QJP ; (v.) P R + R Q- arc P Q is constant; (vi.) the circle inscribed in the region bounded by the straight line AD, the asymptote DT and the hyper bola AQ touches the hyperbola at F\ PT.QV = FK 2 , PY.QZ=c 2 , Qv-PT=QZ, or vZ = PT, sin x cos x cos x sin x cos x cos x 184. The geometrical theorems of 173 for the ellipse hold with slight modification for the mechanical description of con- focal ellipses and hyperbolas from a fixed hyperbola. The threads from the reel must be led round distant points on the hyperbola APQ (fig. 24) and be wrapped on the curve ; and now, starting from F, the confocal ellipse FED will be described, if the threads are led off in the same direction. At D, one thread DT must be supposed of infinite length ; and, beyond D on the ellipse FD, the thread DT must be trans ferred to the other branch of the hyperbola. By making the threads come off the reel in opposite direc tions, the confocal hyperbola DK can be described, starting from D or any other point R. OF THE SECOND AND THIRD KIND. 139 185. The integration of the functions of 77 can now be expressed by means of the elliptic functions, and of the function E am u, defined by E am u =fdii 2 udu. o Then ficstfudu = u E am u o jK 2 GJ^udu = E am u K 2 u. o To integrate a reciprocal function, for instance nd 2 u, we notice that -y- ^ log dn u = K 2 nd 2 u dn 2 u, Cf/U" so that fK 2 nd 2 udu = E am u /c 2 sn u en u/dn u ; o and so on. Again, since cd 2 i6 = sn 2 (j5T u), jK 2 cd 2 udu = u jdu 2 (K u)du = u E + E am ( K u) = u E am u + /c 2 sn u en u/dn u : and since K 2 nd 2 u = dn 2 ^ u), fK 2 nd 2 udu = EE &m(Ku) = E am u /c 2 sn u en u/dn u, as before. In Problem III, 86, we find dt and 7i* =&&QdQ = 0- E am + sn dn 0/cn 0. EXAMPLES. 1. Prove that the area of the Cassinian is 2 / (6 4 a 4 sin 2 <) 2 cZ0, if b > a ; o /~ iT 4 or 27 (a^ o*sm 2 (h)~^b 4 cos~cbdd), if a > 6. 190 THE ELLIPTIC INTEGRALS 2. Rectify, by means of elliptic arcs (pointing out the geometrical connexion), (i.) y/b = sin x/a, cos x/a, cosh x/a, dn x/a, en x/a, sn x/a, . . . ; (ii.) r = bcos(bO/a) or acos(a9/b), the pedals of an epi- or hypo-cycloid ; (iii.) rcos(bO/a) = b, or rcosh(b6/a) = b, Cotes s spirals; (iv.) the Iima9on r = a + bcos0, the trochoid, and the epi- and hypo-trochoids. 3. Express x as a function of s in the Elastica of 97. Prove that if the ordinate is made equal to p, the perpendic ular on the tangent from the centre of an ellipse or hyperbola, and if the abscissa is made equal to the &rcAPPY, the curve will be an Elastica (Maclaurin, Fluxions, 1742.) ,, z . - K v -i. Prove that (1 K 2 )-r^ + -5 -- K=Q: x die K OK d*E \- dE (1 *rhrx+~ j- +E =0. cue K OK Change the independent variable in these differential equa tions from K to k, 6, or u, where K = ^k = sin = tanh u ; and reduce the resulting equations to the canonical form 7 o y dx 2 Solve the differential equations in which 1 _ // I=-7T9jj*> cosec 2 2$, cosech 2 2u, 4Ar/C z (Glaisher, Q. Jl Jf., XX., p. 313 ; Kleiber, Messenger, XVIII., p. 167.) 5. Prove that, if u l -}-u 2 + u 3 -\-u^ 0, 12S4 ll,. 3 x 4 ~ OF THE SECOND AND THIRD KIND. The Elliptic Integral of the Third Kind. 186. We can now make a fresh start, and prove the Addition Theorem for the Zeta Function independently ; and then pro ceed to Jacobi s form of the Third Elliptic Integral. (Fundamenta Nova, 49; Glaisher, Proc. L.M.S. XVII. p. 153.) Multiplying formulas (3) and (6), 137, 4/c 2 sn u en u dn u sn v en v dn v /1N ,..(1) 2 (1-/C 2 S1 and, integrating with respect to v, x n 2 en u dn u/sn it where C is the constant of integration, independent of v. To determine C, first put v = u\ then 2 en u dn u/sn u so that, replacing E am u by Eu/K+Zu, , \ . TJ, \ ryo 2 en u dn tt/sn u 2 en u dn ulsn u Z(u+v)-f-Z(tt-v)-Z2tt= - = - H --- i - 5-^5 ^ 1 /c z sn 4 u 1 rTsnnc. su*v n /csn 1 sn 2 u\ = K -sn(u + v)su(u-v)su2u .......... (2) Replacing u+v t u v, and 2u by u, v, and tt+t?, this becomes the formula given above, 176, Zu + Zv - Z(u -f- v) = /c 2 sn u sn v sn(u + v) ............. (2)* Again, put u = Q for the determination of 0; then C= 2Eu + 2 en u dn u/sn ^ ; and now 2/c 2 sn u en u dn u sn 2 v (3), 1 /c 2 sn 2 w, sn 2 t> another form of the Addition Equation of the Zeta Function, leading immediately to Jacobi s form of the Third Elliptic Integral, as required in 114. 187. Integrating this equation (3) again with respect to v, and employing Jacobi s notation of TT/ \ r /Vsn u en u dn u sn 2 i; dv H(v, u) for / J 1 o where u is called the parameter, and v the argument, then IK 16 = vZu - 192 THE ELLIPTIC INTEGRALS Jacob! now introduces a new function Qu, called the Theta Function, defined by \og~, or so that Now o /Z(u v)dv = log and ,., so that the Third Elliptic Integral is expressed by Jacobi s Theta and Zeta Functions, the arguments being u and v, two in number only, and not three, n, K, </>, as in Legendre s form. 188. Integrating equation (3) again with respect to u, f u f v / 7{dn 2 (t + v} dn 2 (i6 v)}dv du = log(l o o or Q(u v) _, Qu -^-~ ~ 2 lo ~ = , ... log(l - or - = l-^n%sn 2 ^, ......... (6) a formula which takes the place of the Addition Theorem for the Theta Functions. For instance, putting u = v, e%u = (l- /c 2 sn%)e%/6 3 ................ (7) Interchanging the argument and parameter, u and v, then so that II(^, v)-II(v, ^) = uZv-?;Zu, ........................... (8) and Ii(v, u) is thus made to depend upon II(^, v). OF THE SECOND AND THIRD KIND. 193 189. In Legendre s notation, II(7i, /c, 0) or simply 110, is employed to denote his Elliptic Integral of the Third Kind n being called Legendre s parameter ( 114) ; and with Jacobi s notation, H(, K, am u) = But Jacobi changes the notation, by putting n= /c 2 sn 2 a, and by calling a the parameter ; also by denoting the integral V 2 sn a en a dn a snhidu , -,-,- , o and not the integral du ,. , sn a H(u, a) 2 ^ 5-, which equals u-\ /c 2 sn-a sn-n, en a dn a o In Legendre s notation, the Addition Equation of the elliptic integrals of the first kind leads to E<p + E\fr E/JL = /c 2 sin sin i/r sin /A, the Addition Theorem for the second elliptic integrals ; and now for Legendre s elliptic integrals of the third kind, the Addition Theorem is (Legendre, F. E. /., Chap. XVI.) TT , i TT ; TT 1 n*./a sin d> sin \!s sin /m H$ + H\k-U/uL= r tan- 1 - -^_ V a 1 + 71 71 COS <j> COS T/r COS /JL = -J-, tanh-i > /( - a)siD sin t SiPM , (9) ^( a) 1 + n n cos cos \js cos /x according as a is positive or negative, where this can be verified by differentiation. This relation is very much simplified by the use of Jacobi s function II(u, a) ; and now with it becomes H(u, a) + II(v, a) II (u + v, a) = J log fi, where n_B(u-a)9(t;-a)e(u + T; + a) - ............. (1 and 1 is capable of being expressed in a great variety of ways by means of the elliptic functions en, sn, dn of combinations of u, v, a. G.E.F. N 194 THE ELLIPTIC INTEGRALS F /6Q - a)e(v - a)\ 2 _ ~ 80 ~J ~I- K %n*(;u-a)sn(v-a) 00 ~) ~lif /ea9(u + 0-a)\ 2 9(M + fl)9(u + tt-2a) I 00 / ~l- K Wasn\u+v-a) 90" ( 188), so that (Fundamenta Nova, 54) Q 2 _ 1 /c 2 sn 2 (u + a)sn 2 (i> + a) 1 /c 2 sn 2 a sn 2 (t& -f 1; a) _ _ , sn 2 (u + v + a)" One of the simplest expressions, equivalent to that given above in (9) in Legendre s notation, is Q _ 1 it 2 snusn vsna sn(u-\-v a) ,- . l+/c 2 snt(/snvsnasn(u + v + a) " and a systematic collection of different forms of Q is given by Glaisher (Messenger of Mathematics, X.). 190. According as Legendre s or (1 +TI)(! + 2 /^) is positive or negative, so his Integral of the Third Kind LT(X /c, 0) falls into one of two classes, the first called circular, the second logarithmic, or hyperbolic, as we shall call it. In the corresponding classification of Jacobi s form, the para meter a is imaginary or real; and it is remarkable that in dynamical problems, it is the circular form, with imaginary Jacobian parameter a, which is of almost invariable occurrence. When Legendre s a or (l+7i)(l+/c 2 /?i) is positive, and the corresponding Elliptic Integral of the Third Kind is circular, then Jacobi s parameter is imaginary; and (i.) with n positive, we must put n= /c 2 sn 2 m; (ii.) K 2 >n> 1, we must, according to 56, put as in 114 ; and now the integral is expressed by H.(u, id] or H(u, K+ib), involving Theta and Zeta functions of the imaginary arguments ia or K-\-ib ; for which there is no theorem, short of expansion, to express the result in a real form. We shall find however, in the applications, that this imagi nary form constitutes no real practical drawback. OF THE SECOND AND THIRD KIXD. 195 Taking for example the result of 114, then, by (6) 188, with u = nt, and a = K+t iK ; while cm so that, by multiplication, (x + iy)(cos fjit i sin /mt), or p exp i(0 -s/C AC J QuBa which, when resolved into its real and imaginary part, gives the vector of the herpolhode, or its coordinates with respect to axes resolving with constant angular velocity //. 191. Take Jacobi s IL(u, a), and split up the quantity under the sign of integration into a quotient and partial fractions ; therefore Icnadnaf /"" du C du 2 sn a 1^/1 K sn a sn u ,y 1 + K sn a si = u en a dn a/sn a + II(it, a) ; while 1 cnadn a f f du f du :( f- ( J 1 2 sn a t J 1 K sn a sn u J 1 + /c sn a sn 16 J V en a dn a sn u , /csn c sn (a + u) J/c sn(a u) } du /c cn(a + u) dn _ Therefore, by addition and subtraction, en a dn a r du f en a dn a\ I - = u ZcH sna V 1 K sn a sn u \ sna/ o , 1^ Q(a u) dn(a u) /ccn(a 2 t>(a + u)*dn(a+u)+/ccn(a+w) dna /ccna cnadna/" a lt / cnadnaN /r =su(ZaH sn a ^y 1 + /c sn a sn u sn a / . 1, 0(a 11) dn( ^() + /ccn(a u) dna /ccna & 196 THE ELLIPTIC INTEGRALS 192. Again, taking the formula (7), 137, sn 2 a = sn(a + tt)sn(a tt). Ksna snu and differentiating logarithmically with respect to a, Ksn sn _1 cn(a + u)dn(a+u) 1 cn(a u)dn(a u) ~~2 sn(<x + u) 2 su(a-u) and then integrating with respect to u, snacnadnadu 1, sn(o. + i6) S n^-sn% = 2 10g 4^) E(W> ft a-w) 0(o tt) a + .) - ) .............. (14) introducing Jacobi s function Hu, called the Eta Function, defined by the equation (Fundamenta Nova, 61), snu = J_I^ ^/K Bu" This form (14) and Jacobi s II(u, a) are the two forms of the hyperbolic integral of the third kind to which Legendre s form can be reduced for negative values of a. When > n > /c 2 , we put n = /c 2 sn 2 a, and obtain Jacobi s form TL(u, a) of (5). When 1 > n> oo,we put n = l/sn 2 a, and obtain the above form (14). This form again can be split up into partial fractions ; and a similar procedure shows that, since du __, snu , dnu cnu therefore, by equations (4) and (7), 137, cnadnasnudu f 2 ,y sn (a + u) sn( u) , sn(a u) J sn (a + u) OF THE SECOND AND THIRD KIND. 197 cn(g u) dng cng cn(g + u) dna + cng* . -, + cn(a + u) dng cng u) cn(g u) dna + cna ^ Therefore, by addition and subtraction of (14) and (16), /cnadnadu sn g sn u o i 11 0(g+u) dn(g + w)-j-cn(g + it) dng cng G(a u) dn(a u) cn(g u) dna-fcna* /cnadnadu sna+snu ) en (a -h u) dn g + en g (a u) dn(g w) + cn(g ii) dn g en g* By means of equation (6), 188, and the formulas of 123, these relations may be written /cnadnadu sn a sn u o rj -i 2 ^(g + ii) sn ^g en ^(c U/jA-f- iO^ T^TT x U~^(g it) sn ^(a /cngdngfe sn g + sn u o The student may prove, by a similar procedure, that / dnu-dna _ 1 = //c 2 snacna(?-i6 _ , , 1 + dn(a it dnt + dna : 10g T + dH(a+i /snacnadna snucnudnu 7 , sn*a-sn tt / y ciiu-cna = uZ _ 198 THE ELLIPTIC INTEGRAL Eulers Pendulum. 193. Consider for instance the rolling oscillations on a horizontal plane of a body with a cylindrical base, such as a rocking stone, or a cradle. Then the Principle of Energy, considering the line of contact as the instantaneous axis of rotation, leads to the equation JO 2 - 2ch cos + h 2 + k 2 )(d6/dt) = #&( vers a - vers 0) , where denotes the inclination to the vertical of the plane through the axis and the centre of gravity at any time t, a the extreme value of 0, c the radius of the cylindrical surface, h the distance of the C. G. from the axis of the cylinder, and k the radius of gyration about the parallel axis through the C. G. When c = 0, this equation reduces to ordinary pendulum oscillations, as in (3) 3 ; but in the general case we have the oscillations of what is sometimes called Euler s Pendulum. Th d? = {(c- 4tgh cos 2 and now, if we put tan J$ = tan J cos 0, dt on putting 7i 2 = r//c, and 2 _ (c + fe) 2 + fe 2 . /2 _ (c-hY + k* ~c 2 -2c/icosa + A 2 + /c 2S1 ~c 2 -2^cosa + ^ 2 + A: 2C( To reduce this to Jacobi s canonical form, put <j> = &mu, and sin 2 Ja = /c 2 sn 2 a ; then dn 2 a = cos 2 ia, - and sn 2 a = - , y^ -- , cn 2 a = , y^ , ., , . . 2 7g (c + /i) 2 + & 2 (c + /i) 2 + k 2 dt ^snadna dn 2 u so that n = 2 en a, 1 _ sn a dn a _ 2/c 2 sn a en a dn a sn 2 u en a 1 /c 2 sn 2 , _sn a dn a and 7i^=2 u 2II(u, a) en a while OF THE SECOND AND THIRD KIND. 199 In the ordinary pendulum, where c = 0, this reduces, as in 8, to equivalent to sin|$ = sin Ja sn nt ; where n now denotes As another application of the Third Elliptic Integral the student may rectify the inverse (or pedal) of an ellipse or hyperbola, with respect to any point; examining the parti cular case when the point is the centre ; also the case of the Lemniscate, the inverse or pedal of a rectangular hyperbola, with respect to the centre (R. A. Roberts, Integral Calculus, p. 810). EXAMPLES. 1. Prove that, if fc + //=!, (k c / / and deduce Legendres relation of 171. 2 f m f l _ k(y-x)dxdy o o /7w o x^W ^1 -. .7 7. .3 J(l + Kx(^ K y (-x.-(y-l.- K ) (66). (y-x}dxdy _ = ^ -e B )^/(-4 ! .y-e r y-e. 2 .y-e 3 ) ( 51). (y-x)dxdij _ f J ( 47). (/3-a)( 7 -a}(S-a}(y-x)dxdy_ y |8 7. Denoting K-E, K -E , E-K-K, E -^K by J, J , G, G f respectively (Glaisher, Q. J. M., XX.), prove that IK dK \ K f 2 (^dJ T dE\ -5 -- JK.J ) = - 1 JtLi - 7 -- J -j ] d K d K / K\ die die / ( Jf dE r dJ \ 1 ( dG r ,dG\ = K (J -^ -- ~J l = - I VT^J -- ^"^T") V a/c a/c / K \ dK OK/ CHAPTER VII. ELLIPTIC INTEGRALS IN GENERAL, AND THEIR APPLICATIONS. ?. The general algebraical function, the integral of which leads to elliptic integrals, is of the form S+TJX U+ VJX> where S, T, U, V are rational integral algebraical functions of x, and X is of the third or fourth degree in x. We first rationalize the denominator, so that S+TJX__(S+TJX)(U-VJX)_M N 1 U+ vJX ~ U 2 -V*X ~D^D JX 9 suppose ; and now the integration of the rational part M/D is effected by elementary methods, when it is resolved into its quotient and partial fractions. In the irrational part NjD^/X, the rational fraction N/D is also resolved, into a quotient, having a typical term x m , and into partial fractions, having typical terms By differentiation, we find that so that, integrating, and denoting Jx m dx/^/X by u m , x m - 3^/x = (m 1 )em m + 4(m f) bu m _ l + 6 (m + 4(ra - %)du m _ 3 + (m - 3)eu m _ 4 , a formula of reduction by means of which the integral u m is made to depend ultimately on the integrals u 2 , u v and u . 200 ELLIPTIC INTEGRALS IN GENERAL. 201 Similarly, by differentiation and integration, denoting by v m we can determine another formula of reduction, of the form -l = A V n + Bl n - 1 + Cl n _ -2 + Dv n . 3 + El n . 4, / \n {Jb a) by means of which the integral v n is made to depend ultimately on the integrals v v v Qt V-^, and v_ 2 ; or rather, on v v u , u v u 2 ; since r and u are the same, and v -i = u i ~ w o u -2 = U 2 ~ 2ca 1 + a*w . By the various substitutions of Chapter II., u is reduced to Legendre s First Elliptic Integral, while at the same time the integrals u v u. 2 , and i\ are reduced to elliptic integrals of the Second and Third Kind. When x a is a factor of X, the substitution x a = l/y shows that v 1 becomes Jydyj^/T t where Fis a cubic function of y, and v l now reduces to the Second Elliptic Integral. But without carrying out this work in detail, now only of antiquarian interest, we adopt instead the Weierstrassian notation : and by means of the substitutions of the previous chapter we express x and ^/X rationally in terms of pu and p u ; so that the integration is reduced ultimately to that of A+Bp u with respect to u, A and B being rational functions of $11. 195. We must at this stage introduce the functions f i& and <ru, the functions employed by Weierstrass, in conjunction with his function <pu. The function fit, called the zeta function, is defined by &= pu, or f&= Jpudu ; while the function a-u, called the sigma function, is defined by or log a-u =judu, (TIL = expy udu ; and thus - ^ = 9 u du 2 202 ELLIPTIC INTEGRALS IN GENERAL, fi Taking the definition of s or pu in 50, expand in descending powers of s, and integrate ; then the * marking the place of a missing term in the expansion. Therefore, by Keversion of Series, since u 2 is a rational function of s, we obtain, in the neighbourhood of u = 0, To obtain further terms of the expansion, assume $> u = ^2+ * + c l u 2 + c^ + c 3 u Q +...+c n and since > 2 u = 4? 3 u we can obtain from the last equation a recurring formula for the determination of the coefficients c ; and as far as U B , The expansion of the zeta function is now fu = + -^ 3 -^!_ #2 V u^ 60 140 2 4 .3.5 2 .7 2 4 . 3. 5. 7. 11 so that, defined more strictly, o Similarly we shall find, for the sigma function, f-f~/JI ; A I - |^ *-7 *J 3 ^_ IS 2 ~ * 2 4 .3.5 2 3 .3.5.7 2 9 .3 2 .5.7 2 7 .3 2 .5 2 .7.11 so that, strictly defined, loc <ru = log u + / ( tu }du, or <jii = ^exp / \(u )du. J \* u/ v t/ V s u/ AND THEIR APPLICATIONS. 203 Homogeneity. 196. From considerations of homogeneity it follows, that if u is changed into u/m, and at the same time if g. 2 and g 3 are changed into ??i 4 <7 2 and m 6 (/ 3 , then s or <pu is changed into m 2 s or m 2 $w ; so that }- 1 ( Vj lit \ttir . \_ i-y fOL. and similarly _ 1 i(u a{u ; g.-,, g B ) = m <r ( ; At the same time the discriminant A becomes changed to m 12 A, but the absolute invariant J is left unchanged ( 53) ; we may in this manner alter the argument u proportionally ; for instance by taking m = i/(e 1 - c 3 ) we can make the argument the same as in the corresponding elliptic functions ( 51). When m is chosen so that m 12 A = l, or m=A"*,the elliptic integral is said to be normalised (Klein). Suppose, for instance, that g = 0, and m, m 2 are the imaginary cube roots of unity, J Ji^/3 ; then m 3 = l, and u/m = m z v,-, so that $p(m 2 it ; 0, g^ = m?<p(u ; 0, g 3 ), p(m u ; 0, g s ) = m f<u ; 0, g s ), 1111(3 w iC ~-~ v) iit/tL O if1>~ / lJlf* <r(u 0, f/o) = m<r = m~v L/^/ /yyj />T1 This is the simplest illustration of the theory of Complex Multiplication of Elliptic Functions, of which we shall make use hereafter ; the general theory is required in the integration of the equation Mdy dx for particular numerical values of g. 2 and </ 3 , when l/M is a complex number of the form a + ib^/n ; in this instance g. 2 = Q> and M is an imaginary cubic root of unity. 204 ELLIPTIC INTEGRALS IN GENERAL, 197. With the aid of these three functions of Weierstrass^ <pu, fit, and oru, it is possible to express any elliptic integral, and we can thus complete the problem left unfinished in 194. The function f u is analogous to Jacobi s Zeta function ; and with s = <@u } it may be defined by the relation & =J S -jji =/(4s 3 - 2 s - </ 3 ) - *s ds * Thus, for instance, from 153, with appropriate limits, . as-0-3 dx i \a /3 a y a where u=f^=. J sj -& To obtain the Addition Equation of the zeta function analogous to (2) and (3) of 186, take the formula (F) of 144, it ll, ft/? A ^ implying also the formula, obtained by changing the sign of v, ftt+r so that, by subtraction, , (u _ v) Integrating (a) with respect to v, where C, the arbitrary constant of integration, may be obtained by putting v = 0, when p; = oo ; so that 0= 2u, and An interchange of it and v gives -8-)+ so that, by addition, (y) the Addition Equation, analogous to (2*) 186. AND THEIR APPLICATIONS. 205 With U + V + W = Q ) this may be written, analogous to 176, 198. We can now take the function A+Bp u of 194, and suppose that A and B are resolved into their quotient and partial fractions. Writing p, p , p", ... for <pu and its successive derivatives, then the relations pt= 4^ 3 -#,p-(/ 3 p"= W-to p" =l2pp\ etc., enable us to express the quotient or integral part of A + B p u in the form C = C Considering next a partial fraction of A + B y u of the form we replace a by pi 1 , and write the partial fraction in the form pu-pv All such partial fractions can thus be expressed by a series of terms, where the sum of the coefficients I is zero for each partial fraction, and therefore for the whole series ; so that ? 1 + / 2 + / 3 +...=0. Again, by repeated differentiation of equations (/3) and (ft ) ( 197), with respect to u or v, we obtain equations, such as by means of which partial fractions of the form P + Q& u .. P + Qy u ,o, or generallv . . u w* 1 J * can be expressed by terms of the form ^(u + r), $>(u v), and by their derivatives ; as well as by terms of the form L and C. 206 ELLIPTIC INTEGRALS IN GENERAL, Thus, finally, A+Bp u, or any rational function of <pu and p u, can always be expressed as the sum L + P of two series of terms, L = l(u - vj + l(u - v z ) + l B {(u - r s ) + . . . , where ^ + Z 2 _|_ g + . . . = Q, and P = c + 2m ^(u v) ; and now the integral can immediately be written down, in volving, in general, the sigma, zeta, and $ function, as well as its derivatives. When the sigma and zeta functions are absent, the integral is a function of @u and p u, and is not properly elliptic, but only algebraical. This method of integration is taken from Halphen s Fonc- tions Elliptiques, L, chap. vii. Halphen points out that to obtain the coefficients in the series of terms I flu -v)-}- m$(u -v) + m^\u -v} + m.$"(u - v) + . . . , corresponding to the same v, it is only necessary to take the coefficients of (u v)-\ (u-v)~ 2 , (u-v)~ 3 , ... in the expansion of A+B$ u in ascending powers of u v; the coefficient I being Cauchy s residue. 199. Integrating (j3) with respect to v, then ff udv^ ^uy) ..... J <$U-<$V *(r(U-v) S lPl which may be considered a canonical form of the Third Elliptic Integral, in Weierstrass s notation. Thus, for instance, in 113, By integration of (y), with respect to u and v, I i u i v- -, /I ip u % pU /I W U Q VJ tZv = 2 ^-^v AXD THEIR APPLICATIONS. 207 either of which may be taken as a canonical form of the Third Elliptic Integral ; and also as illustrating the interchange of amplitude u and parameter v, as in the Jacobian Elliptic Integral of the Third Kind, TL(u, v), in 188. Or otherwise, interchanging u and v in (fa), or integrating (/3 )> so that, by addition of (fa) and (fa), . -.(<?) a form of the theorem of the interchange of amplitude and parameter, analogous to (8), 188. 200. Integrating (ft) with respect to u, the fundamental formula is the use of Weierstrass s elliptic function, analogous to equation (6) of 188. As an application consider the herpolhode of 113 ; then (jUoV while #* = . l^ u + v ( e ~ t \<K*-t;) so that, in the curve described by H, fo + t a-ua-v while in the herpolhode described by P we must multiply this function by e iflt or cos /mt + i sin jmt. Putting u = v in (K), we obtain (7 (7 This may be obtained by integration of the formula of 149, 1 d 2 208 ELLIPTIC INTEGRALS IN GENERAL, If u, v, w, x denote any four arguments, <r(u - v )<r( u + v )ar(w - x)a-(w + x) 0, ............ (L) since it is of the form where U- V=- ^Ua^v^u - QV), etc. 201. We notice that the Third Elliptic Integral can be expressed very simply as the logarithm of a function, so that we may write (y^ in the form uv ft/ - -v- ii V-*V ) ^ ) ) o 7 i , . o-(u-\-v] } where ^C^, v) = -&-*** eru arv and </>(u, v) is called by Hermite a doubly periodic function of the second kind. Changing the sign of u, or v, (jU (TV so that <p(u, v)<p(u, v) = <@u $>v. 202. Suppose @v = e lf e 2 , or e 3 ; then, according to 54, we can take v = u> v ^ + 0)3, or w 3 , to correspond; and now <p v = 0, and log (j>(u, v) = % log ($>u pv) ; so that 0(u, wj) = <f>(u, - Wl ) = J($>u - ej, etc. ; and </>(u, v) is an elliptic function for these values of v. We may thus put , or ^, where a-.u denotes Similarly, where Also ^>% = - 2^/($>u -e r pu-e. 2 . <pu -e 3 ) = - Z^u cr. 2 u ar 3 u/a*ii, and ( 200) AND THEIR APPLICATIONS. 9Q9 Denoting by a, /3, y the three numbers 1, 2, 3, taken in any order, then the relation gives, by a combination of the expansions of o~u and <pu in 195, so that o- a u is an even function of u, and unaffected by Homo geneity ( 196). Thus, for instance, from ex. 9, p. 174, >2u - e a The symbol ^ is employed to denote fa) a , so that r\ is the analogue of Legendre s E of 77. With positive discriminant A ( 53), we find (exs. 4, o, p. 199), and with negative A ( 62), formulas analogous to Legendre s relation of 171. 203. Denoting $m, pv, pie; by x t y, z, then ( 165) if u + v + w = 0, (x^y + z}(^xyz-g z ) = (yz + zx + xy + lg^ ................. (I.) Denoting also (x ea)(y e a }(z e a ) by s a 2 , then since a + (a; __zzx x 2x ze by means of (I) ; and this is of the form A+Be a , so that (e, - g s )s i + 3 - ei )s. 2 + (c t - e 2 )s 3 = ; or (e z - e 3 V 1 uor 1 i o-^+(e 3 - e 1 )o- 2 ito- 2 7;o- 2 w + (e 1 - 62)0-3160-3^0-3^ = 0, snce (W. Burnside, Messenger of Mathematics, Oct. 1891.) As an exercise the student may prove that, with u + v + iv + x = 0, (e. 2 e a )criU (r-^v 2 - 3 a - 11 - the analogue, in Weierstrass s notation, to Cayley s theorem, given in ex. 1, ii., p. 140. G.E.F. o 210 ELLIPTIC INTEGRALS IN GENERAL, 204. The solution of Lamp s differential equation, which may be written in Weierstrass s notation .(1) is given, when n = l, by the function 0(u, v) of 201. For, differentiating < logarithmically with respect to u, 1 d(b 1 & u & v ,. N c c -- 5*-=5- - =(u + v) tu tv, $ du 2 ^u-^^ and differentiating again, 1 d 2 <f> I d<{> 2 , <j>du*-fdu* = so that du 2 4 Lame s differential equation, with n = l, and h = $v. The general solution of is therefore y = C<j>(u, v) + C (f>(u, v), or C<t>(u, v) + C <p( u, v). When h or @v = e lt e 2 , or e 3 , the solution is one of Lamp s functions, as in 202. One solution is now *J($u e a ), where a = l, 2, or 3; the other being { f (u + co a ) e a u} /J^u e a ) t as may be verified by differentiation, or determined indepen dently from a knowledge of the particular solution ^/(<@u e a ). 205. The revolving chain, resumed. We are now able to complete the solution ( 80) of the tortuous revolving chain, by obtaining an analytical expression for its projection on a plane perpendicular to the axis of revolution. Putting y = r cos i/r, z = r sin i/r, then we have found in 80, p. 70, that, when the notation of Legendre and Jacobi is employed, d^_H H/T dx ~ Tr* AND THEIR APPLICATIONS. 211 which, on putting u = Kx/a, and so that, with K 2 = (6 2 - c 2 )/(cZ 2 - c 2 ), sn 2 v = - (d 2 - c 2 )/c 2 , cn 2 ^ = c? 2 /c 2 , i cZi\/r cn v dn v/sn v becomes ~f- = - - ^^ -- , ft i& 1 trSDrli sn v , , . , cn t dn i; ,-,, x ,, x so that i\/s=u- --- IL(u, v) ...................... (1) Since sn 2 y is negative, we may, by (67) 73, put v = t iK , where t is a real proper fraction. Now r = =ceo / \ -,., .$ /9(u-fv) / cnvdnv \ while e l V=J 7 -- (exp --- -- Zv)u; \6(u y) snv / j.v w^ 0(^ + v) / cn v dn v \ so that y + iz = cOQ -^^ exp -- -- Zv)u:....(3) 016 Ot; sn v which, when resolved into its real and imaginary part, will give y and as functions of u or Kxja, and thus represent the equation of the chain. y ^ 206. The procedure is more rapid with Weierstrass s notation. Writing 2/ 2 + z 2 = r 2 , we have found that ( 80) ( r 7.,2\2 /4, / ,2 =^o--^+^-c), so that we may put - S w), ..................................... (1) j j J.V j. provided that ~-i-= and g 2 , g% are suitably chosen. Since v is the value of u which makes r 2 vanish, therefore the value of (dr 2 /dx) 2 when r 2 = ( 80) ; so that ^ 2 y=: -16lT 2 /?i% 2 ^ 6 , ............................... (2) and <p v is therefore a pure imaginary, which we take to be negative imaginary, so that v = t w 3 ( 54). Now d = H_ dx^ -IE 1 )&v_ dn Tr 2 du n-wJ& pu pv %>u-$>v 212 ELLIPTIC INTEGRALS IN GENERAL, or = --- = b&v + u) + tf(v-u)-tv ...... (3) du %>upv Jiv from (ft) ( 197) ; so that (i> tt) while r - k = ................... (5) - 2 * and + fe= fc crVcrU = Jc</>(u, v), y-iz = k<f>(u,-v), .................................. (6) giving the form of the chain. For a revolving chain fixed at two points, we must have r 2 restricted to lie between positive values, 6 2 and c 2 , and therefore $u must be restricted to lie between e 2 and e 3 ; so that with du/dx constant, we must put u = x For a chain attracted to the axis with intensity proportional to the distance, and thus taking up a form of minimum moment of inertia, we have u = xcvja ; and now pu can become infinite, and the chain reach to infinite distance. In this and other mechanical problems, the parameter of the elliptic integral of the third kind is almost always imaginary ; the apparent awkwardness of this imaginary parameter is removed when we proceed to express the vector y-\-iz by a doubly periodic function of the second kind $(11,, v), whose logarithm is the elliptic integral of the third kind ; and thence determine y and z theoretically by resolving </>(u, v) into its real and imaginary part. Familiar instances of the same procedure are met with in Elementary Mathematics ; thus x + iy = c cos(nt + ia), or c cosh(nt + i/3), will represent elliptic or hyperbolic motion about the centre. Generally, with x -f iy z, X+iY=Z=F z: then will give the motion of a particle of unit mass under component forces (X, F). (Lecornu, Comptes Rendus, t. 101, p. 1244.) AND THEIR APPLICATIONS. 21:3 207. The Tortuous Elastica. A procedure, similar to that just employed for the revolving chain, will show that the equation of the curve assumed by a round wire of uniform flexibility in all directions can be expressed by the equation y + iz = k<l>(u, v) and z = ku + yii, where u = swjc + w 3 , s denoting the length of an arc of the wire, and 2c the length of a complete wave. (Proc. London Math. Society, XVIIL, p. 277.) The elastic wire differs thus from the revolving chain in having it = so^/c + w s , instead of tt=a^ 1 /a-hw 8 ( 97). To establish these equations, take the axis Ox as the axis of the applied wrench, consisting of a force A^ along Ox and a couple L in a plane perpendicular to Ox ; denote the tor- sional couple about the tangent at any point by G, and the flexural rigidity of the wire by B. Then the component couples of resilience about the axes Ox, Oy, Oz are taken to be B(y z"-y"z), B(z x" -z"x ), B(x y"-x"y ) the accents denoting differentiation with respect to the arc s ; the equations of equilibrium are therefore B(yz"-y"z }=Gx+L (1) B(z x"-z"x) = Gy + Xz (2) B(xy"-x"y }=Gz-Xy (3) (Binet and Wantzel, Comptes Rendus, 1844). Differentiating each equation with respect to s, multiplying respectively by x, y, z, and adding, gives G = ; so that G is constant. Multiply equations (1), (2), (3) by x, y, z\ and add ; then -r L* "^- X(yz - y z) = 0, so that yz y z = r 2 d\fs/ds = G/X, a constant ; and yz" y"z = 0. Again, multiplying (2) by y, (3) by ~, and adding, gives Bx\yz - y z} - Bx (yz" - y"z) = G(yy f + zz \ or Bx" = X(yy + zz \ so that, integrating, Bx = ^X(y- + 2 ) + H. 214 ELLIPTIC INTEGRALS IN GENERAL, Then = 2X(Bx -H)(I-x 2 )-G 2 , a cubic function of x ; so that, by inversion of the elliptic integral, x or y 2 +z 2 is an elliptic function of the arc s, which may be written y 2 + ^ = & 8 (pa,-pw), ........................... (4) or Bx . , , du provided A dx and now du also = iG ds = 2iBG 1 = frfa ^ 23 By KirchhofFs Kinetic Analogue, it follows that the axis of a Spherical Pendulum, Gyrostat, or Top can be made to follow in direction the tangent of a certain Tortuous Elastica, when the point of contact of the tangent on the elastica moves with constant velocity ; so that, if x, y, z are the coordinates of a point fixed in the axis of the Gyrostat, and Ox is vertical, 7 d o-(u + a)) , . . = k- r - -exp(X MU, du cru o-a) where now u = ^ + o) 3 , and 2 1 /w is the period of the oscillations of the Top, or Spheri cal Pendulum. The Spherical Pendulum and the Top. 208. To prove these formulas independently for the spheri cal pendulum, let the weight of the bob be W lb., and let the tension of the thread be a force of Nl W poundals ; then the equations of motion are, with the axis of x drawn vertically dovmwards, d 2 x d 2 d 2 z A /1X 0; ......... (1) subject to the condition, I denoting the length of the thread, AND THEIR APPLICATIONS. 215 The equation of energy is (2) while yz yz = /^, a constant ................ (3) Now, xic + T/i/ + 22 4- ^ 2 = ##, so that JV7 2 = </ + x 2 + # 2 + z 2 = g(3x + 2c) ; thus giving the tension of the thread. Hermite writes (Sur quelques applications des fonctions elliptiques, 1885) (y + iz)(y - iz) = yy + zz- i(yz - yz) = xxih, so that the norm of each side is (y 2 +z*)(y z +z*) = xW+h\ Then (V-x z ){2g(x + c)-x 2 } =xW+h?, or I 2 x 2 = 2g(x + c)(l 2 - x 2 ) - h 2 so that x is a simple elliptic function of t, which we may write x = k(pv-&u), ........................ (4) where u = nt+a)^ for ^16 to lie between e. 2 and e s . Then l 2 k-,t Y 2 ii = 2gJ^u - pvf - 2gck 2 (pu - pv)* - pi ) + 2gd* - h 2 provided n 2 = ^gkjl 2 , and py = Jc/& ; while ^ 2 and g 3 are suitably chosen. The value of y v is found by noticing that x = when u v; and thus l 2 k 2 nY 2 v = 2gcl 2 - h 2 , Now Hermite writes d 2 . j-5(3/2f= -r- 9 = -- 1- = , ^ 2V< c/it 2 grfc A; Lame s differential equation for n = 2, with ^=6|w. The formal solution of this equation is reserved for the present; but it can be inferred for this case by taking the equation (3) and writing it ^ = h du n(y 2 + z 2 ) di\ts_ ih/n _^i 216 ELLIPTIC INTEGRALS IN GENERAL, We now put ......... (6) and since I 2 x* = h z , when x = I, or when u = a, or 6, therefore 79 With & positive, and 06>0u>0a, we take p a negative imaginary, and 6 = a positive imaginary, so that ( 54), , where p and q are real proper fractions. Then = - +s_. ... (7) au 016 0a <pb $u and integrating, by equation (/3), 199, i 11 <r( U + a) . ., cr(6 + u) c7 /Q \ V =l 1 g^=r ) -f+i 1 8-^-( 8 > Now while j.i , 7 ,. - so that v + ^ = ^ - exp( - fa - era 0-6 9% -,cr(u a)a-(b u /nx (9) thus giving the solution of Lame s differential equation for n 2. 209. It is interesting to verify that these values of y + iz and y iz are solutions of Lame s equation for n = 2. Denoting y-\-iz by 0, and differentiating logarithmically, and differentiating again, 1 p u-p a 1 p b-p u. 2 u wt 2 <b <u l/0 i6 a\ 2 . 1 <QU a 6 0V l/0 6 /r i = . I -5 ) ""~ 9 ^^ 1 + 4^7; -K6+^) AND THEIR APPLICATIONS. 217 u &a p 6 p u a- T 2 pu pa po*-fne. But with p a = p 6, tt p a p 6 & u - - \ - pit) -rt = 6pit -{- 3pa + 3p6, <h CiU Lame s differential equation for ?i = 2, with 7t = 3pa + 3p6, in place of the previous value of h = 6pv. From KirchhofF s Kinetic Analogue in 207 we may put -, N ex P( ft> tyu era orb cr d I where X = f (a + b) fa f 6. With p (a - 6) = p a = p 6, therefore f (a 6) = fa f 6 ; and, changing the sign of a, P7 cZ exp(ca Co)u=-j-A(tt, a ^ 7 9 o-a 0-6 a- 2 u (Halphen, F. E., I., p. 230.) 210. In the slightly more general case of the motion of the Top, we shall find it convenient to draw the axis Ox vertically upwards, and to call the angle which the axis OC of the top makes with the vertical Ox. Then, from the principles of the Conservation of Energy and Momentum, we obtain the equations (Routh, Rigid Dynamics) i A (dO/dty + \A sm 2 0(aV/cZ0 2 = Wg(c - h cos 0) , ...... ( 1 ) ^sin 2 0(^/^0 + ^ cos0=^, ...................... (2) where r denotes the constant angular velocity of the top about its axis of figure OC, d\frjdt the angular velocity of the verti cal plane through Ox and OC, h the distance of the centre of gravity G from 0, W Ib. the weight of the top, and C, A its moments of inertia about the axis of figure OC } and about any axis through at right angles to OC. 218 ELLIPTIC INTEGRALS IN GENERAL, Putting A/Wh = l=OP, as in the simple pendulum, then P is the centre of oscillation for plane vibrations. The elimination of d\fsldt between equations (1) and (2) gives = (/(cos cos a)(cos cos /3)(cos d), ...... (3) suppose ; the inclination of the axis of the top to the vertical being supposed to oscillate between a and /3, a > > /3, or cos a < cos < cos /3 < d. Guided by equation (17), p. 37, we put cos = cos a cos 2 < + cos /3 sin 2 <, cos cos a = (cos /3 cos a)sin 2 <, cos/3 cos 6 = (cos /3 cos a)cos 2 < ; ................ (4) and therefore, = o j{d cos a (cos/3 eosa)sin 2 </>} , 9 cos 8 cos a , 9 d cos ^8 where /r = -- s " K ^ d cos a a cos a and in- 2 = J </(c? cos a). Now we may put < = am nt, and cos = cos a cn 2 ?i + cos /3 sn 2 7i, ................ (5) so that the projection on the vertical Ox of the motion of a point on OC resembles ordinary plane pendulum motion. When d = 1 and cos a= 1, then G and Cr vanish, and the oscillations are in a vertical plane. But, in the general state of motion, ._ dt = "sin = 1 G + Cr 1 0- 2 1 G^ + Or 1 _ G-Or _ 2 l-cosa-(cos/3-cosa)sn 2 ft ~2 so that \/s is expressed by two Third Elliptic Integrals. AND THEIR APPLICATIONS. 219 Putting cos#= 1 in equation (3), show that = = 2(l +cos a cos = 2(l -cos a)(l -cos /3)((Z- 1), i while, in accordance with Jacobi s notation, we put cos 3 cos a 9 cos / cos a . = , 1 + cosa 1 cos a so that, finally, with u = nt, we find en t\n vsn. v t . en and, as in the spherical pendulum ( 208), we take v l = ipK , r. 2 = K+ iqK , where p and q are real proper fractions. In the Weierstrassian notation, we put, as in (6), 208, 1 + cos = k($ni pa), 1 cos 6 = and thence ( 22-i) c - h cos = hk{p(a + b)- ^1-1- AT, V A , We thus obtain -5-*-= --- 1 du pu $>a but now the relation p a= p & holds only when 0;- = 0, or when the motion of the top is comparable with that of the spherical pendulum ; on the other hand, the relation p a = $ b implies that G = 0. The Kinetic Analogue of the Top with the Tortuous Elastica ( 207) is obtained by putting a + b = a), and X = f(a + 6) a f&. In the Steady Motion of the Top, a = /3, K = 0, K= JTT ; and the elliptic functions degenerate into circular functions. We thus obtain the condition for the steady motion, and the period of the small oscillations, given in Ifoutb s Rigid Dynamics. 211. A similar procedure will solve the general equations of motion of a solid figure of revolution, moving under no forces through an infinitely extended incompressible friction- less liquid ; the work will be found in Appendix III. of Basset s Hydrodynamics, vol. I ; also in Halphen s Fonctions elliptiques, II., chap. IV. The problem is of practical interest from its bearing upon the determination of the amount of spin requisite to secure the stability of an elongated projectile. (Proceedings, Royal Artillery Institution, 1879.) 220 ELLIPTIC INTEGRALS IN GENERAL, 212. We again resume the consideration of the motion of a body under no forces, first mentioned in 32, as affording a good practical illustration of the necessity for the introduction of various analytical theorems of Elliptic Functions. Geometrical Representation of the Motion of a Body under No Forces, according to MacCullagh, Siacci, and Gebbia. Quadrics concyclic with the moinental ellipsoid, that is, having the same circular sections, are given by (Smith, Solid Geometry, 170) (A -H)x 2 + (B- H)y 2 + (C- H)z 2 = Dk 2 ; and now, if we produce the instantaneous axis of rotation OP to meet the concyclic quadric in P , and denote OP by R, (A -H)p* + (B- H)q 2 + (C- H)r 2 = DAV/E 2 , while Ap* + Bq 2 + Cr^Dh so that, by subtraction, M 2 R 2 J C * R 2 R 2 ~ D Along the polhode, R h sec 6, where denotes the angle between the instantaneous axis OP and the fixed axis of resultant angular momentum 00 ; and then W = COS 2 6 _ H (l} the polar equation of a quadric surface of revolution. Since R 2 is less than A 2 sec 2 $ for all points adjacent to P on the momental ellipsoid, therefore in the concyclic quadric 1 . cos 2 H -f^ 9 is greater than except at the point P , and therefore the concyclic quadric touches this quadric surface of revolution at P and rolls upon it during the motion. We may also take concyclic quadrics, given by and now __ C0 ^, ..................... (2) the polar equation of a quadric of revolution. In particular, if H = D, then .K sin 6 = h, the polar equation of a cylinder of revolution, outside which this concyclic hyper- boloid rolls during the motion (Siacci, In memoriam D. Cheliniy Collectanea mathematica, 1881.) AXD THEIR APPLICATIONS. 221 213. By reciprocation of these theorems, we prove Mac- Cullagh s theorem, " that the ellipsoid of gyration, ^ , f, z*_ I A^B^C M* always moves in contact with two fixed points on the axis of resultant angular momentum, equidistant from the centre " ; and we also deduce Gebbia s extension of MacCullagh s theorem, that " conf ocals of the ellipsoid of gyration, the polar recipro cals of the concyclic ellipsoids of the momental ellipsoid, slide without rolling on fixed quadric surfaces of revolution." In particular, the polar reciprocal of Siacci s cylinder of revolution is a circle, upon which a certain confocal to the ellipsoid of gyration slides without rolling. Geometrical Representation of the Motion, according to Sylvester, Darboux, omd Mannheim. 214. In Sylvester s splendid generalization of Poinsot s re presentation of the motion of the body, it is proved that a confocal to the momental ellipsoid rolls upon a plane per pendicular to the axis of resultant angular momentum OC at a constant distance from 0, which plane rotates about OC with constant angular velocity, and therefore gives a geometrical representation of the time. (Phil. Trans., 1866.) The proof of this theorem depends upon tw r o geometrical propositions, in connexion with confocal quadric surfaces (i.) "The locus of the pole of a fixed tangent plane to a quadric surface, with respect to any confocal, is the normal to the first surface ; " (ii.) " the difference of the squares of the perpendiculars from the centre on two parallel tangent planes of two confocals is constant and equal to the difference of the squares of the corresponding semi-axes." Thus, in fig. 25, if OP is a surface confocal with the momental ellipsoid OP, then Q, the pole of the invariable plane CP with respect to the surface OP , will lie in the normal PQ to the momental ellipsoid at P ; while the surface OP will touch a plane C P , parallel to the invariable plane CP, and such that OC - = OC 2 -\~, X 2 denoting the difference of the squares of corresponding semi-axes of the confocals. 222 ELLIPTIC INTEGRALS IN GENERAL, Since C is a fixed point during the motion of the body, therefore C" is also fixed. Drawing the plane QL through Q, parallel to the invariable plane, and denoting OC by h, as before ; then since Q is the pole of OP, OQ.OV=OP" 2 , or OL.OC= so that OL = h- \ 2 /h, LC= Fig. 25. Again, denoting as before ( 104) by /JL the constant com ponent of the angular velocity of the body about OC, so that the resultant angular velocity of the body about OP is m cosec OPC, then the velocity of the point P in the body is p cosec OPC . OP . sin POP = ^.P V, where V is the point in which the line OP cuts the plane C P . Therefore the angular velocity of P about the invariable r nn . p v py PQ A 2 line OU is /x (7 7 P 7 = M OP =M 00 :=/x /t 2 a constant ; so that if the surface OP rolls without slipping on the plane C P , this plane must revolve about OC with constant angular velocity yuX 2 //i 2 . The point P lies in the plane OQPC ; and since C P _C P _OC f _OC CP~ LQ -OL-~OC" therefore OC . C P = OC . CP, and P lies on the rectangular hyperbola PP ; this is the geometrical property principally employed by Prof. Sylvester. (Solid Geometry, Salmon, 167, 180; Smith, 163, 167.) AND THEIR APPLICATIONS. 223 The angular velocity of the vector C P with respect to the revolving plane C P being ~7?"~Mp & follows that, if p, $ denote the polar coordinates of a point P on the herpolhode described by P on the revolving plane C P , then n 2 OC" 2 K , _ X 2 \ A-D.B-D.C-D equations similar to those required for the herpolhode of P. In particular, if we take \ 2 = k 2 , then 00 = 0, and the con- focal OP is a cone ; and the plane through rotates with constant angular velocity /UL, while the cone, called by Poinsot the rolling and slipping cone, rolls on this revolving plane, the angular velocity about the line of contact OH being v. If we consider the curve described on this revolving plane by the point H, the foot of the perpendicular from P on the plane, then p, being the polar coordinates of H ( 113), dt ~dt ABG~ ~ ~p so that the point H describes on the revolving plane an orbit as if attracted to ; and, as in 89, we shall find that the requisite central force is of the form Ap + Bp s . (Pinczon, Comptes Rendus, April, 1887.) This is otherwise evident, by noticing that the vector x+iy of this curve satisfies Lame s equation ( 204) 72 jj^x + iy) = (2$m + &v)(x + iy), where so that % A value of X may be found which makes the herpolhode of P a closed curve ; and this closed polhode is an algebraical curve, when v is an aliquot part of a period, the correspond ing elliptic integrals of the third kind becoming pseudo-elliptic. Abel has devoted great attention to the subject of pseudo- elliptic integrals ((Euvres, XL), and the algebraical herpolhode affords an interesting application of his theorems ( 218). ELLIPTIC INTEGRALS IN GENERAL, The Addition Theorem for the Third Elliptic Integral. A 215. Theorems (9) and (10) of 189 show that, employing the function $(11, v) of 201, log or or i ai 2 = o o-^ + ^ -h te. 2 )orv a-u^ju^ where, expressed by elliptic functions of u v u 2 , and v , U ~ V + U Also, as in equation (8), 188, log (fr(v, u) = log 0(u, v) + i^fy vf tfc ; so that log 00, O + log^ , u 2 ) = log <f>(v, Ul + u 2 ) - { fa + fu. 2 - ^ + u. 2 ) } v + log Q, . . . (3) the Addition Theorem for the parameters u v u 2 . These theorems have been generalized by Abel for the addi tion of any number of amplitudes or parameters in the Third Elliptic Integral, and the proof is a simple extension of his method, employed in 162 (CEuvres, XXL). Denoting by a any arbitrary quantity, equation (7) of 162 may be written 1 dx r Ox r a x r *JX, r ~~ (a x r )\j/x r Now, since Qa is of lower degree in a than \fsa, and it follows that, when resolved into partial fractions, 6a _ y 6x r and therefore, writing fx and <px for P and Q respectively, and A for the value of X when x = a, ^, 1 dx r 6 a 9 <f>aSfa faS</>a = V^ = (fa) 2 - *J a fa + <t>a or 2 AND THEIR APPLICATIONS. 225 Integrating, with the notation ( 197, 199), so that <r(v+U r ) n- < is expressible by elliptic functions, ? and #> , of v ; provided that, as in (11), 162, t m = 0, or 2tt- r = 2u r , (6) the coefficients in fa and <pa being determined as functions of pu r and $ u r by the plexus of equations (4) in 162 ; fa and <p a being the same functions of u r . Thus the function is an elliptic function of v provided that the sum of the values u r ofv which make the function vanish is equal to the sum of the values u r which make the function infinite ; in other words, briefly expressed, provided the sum of the zeroes u is equal to the sum of the infinities u . In particular, with the u r s all zero, 1,u r = ; and in equation (6), 162, we can put \fsa = (fa) 2 so that 2 log <p(u r) v) = log(fa + </>a . *JA ) + constant. Thus 11^,.,*), or ^+%)^+Xl_-HO ) ....... (8) when u l + u 2 -{-u 3 +...+Uf Jll = 0, ..................... (9) is a rational integral function of $>v and $> v, which may be written, as in 198, v .............. (10) G.E.F P 226 ELLIPTIC INTEGRALS IN GENERAL, So also, since ( 201) therefore, writing 7" for %*, r=ui-l 2 log (j)( U r> v) = log 0( U, v) + log Q + a constant, ...... (11) r=l where Q = C f /(p 17 #w). In particular, when 7=e a , 0(#, v) = ^/(pv-e a ) ( 202), and r n"J<tt r ,t;)=a/V(fw-e tt ) > .................. (12) ? = ! when u 1 + u 2 + u 3 +... + u f ji-i = (ji) a . By an interchange of amplitude and parameter, 2 log (f>(u, v r ) 2 log 0(i&, yV) = log Q pu, .......... (13) provided that 2^ = 2^ 12 being a function of $m, ^ u, $v t $> v ; and p = 2(fy r -f y / r). . 216. A further application of Abel s Theorem of 162 shows that p is expressible as a function of pv and $ v\ this is the generalization of the Addition Theorem for the Second Elliptic Integral, given in 186. For and this case can be determined as a degenerate case of the preceding result ; since, making a oo , Ja-X T JX T = the coefficient of I/a 2 in the expansion in ascending powers 1 . fa da. ,JA /-, 1X of I/a of -^ log^^j ...................... 0- Thus, with Z = 4fl3 3 - # 2 a - ^/ 3 , and a; = py, then and p Jacobi calls V^ the /^c^o? o/ tae 27iird Elliptic Integral (Werke, II., p. 494.) AND THEIR APPLICATIONS. 227 217. Similar results hold when, as in 167, X is supposed resolved into two factors, X l and X. Denoting P^ - Q*X 2 by ^x y and varying the arbitrary coefficients in P and Q, and conse quently the roots of \^x = 0, as in 162, then ^x r dx r + 2PSP . X - 2QSQ . X, = 0, while PJX^+QJX^Q , so that ^ x r dx r - 2(Q3P - PSQ^X^) = 0, or far = 2 Q-dP-PSQ^ Ox r ^/X r \]s x r \ls x r and I,dx r /^/X r = 0, or 2u r = 2u r . Again, as in 215, 2_L_ dx r _ Oc<^ _ a-x r ~ ~ ^ loo , * Thus, as an application to the formulas of 174, 176, 186, and 189, take, as in 38 (Durege, Elliptische Functionen, 36), X = X 1 X 2 , where X^ = x, JT 2 = (1 #)(! kx). Then, with x = sn 2 u, c rxdx 2 , and ; in Legendre s notation, with = arn i^,, and ??-== I/a. Now, if, as in 164, 165, we take P or ixp-\-x, and Q or (f>x = q, and denote by x v x 2 , x 3 , the roots of the equation (7), 167, V*, or P^-Q 2 ^, or then x l + so that, as in 164, (2 - ^ - ;r 2 - a; 8 + kx^x^ = 4(1-^.1-^.1- a; 3 ) f where 228 ELLIPTIC INTEGEALS IN GENERAL, AgaiD, a dx r a ^Xt, , f a a-x l ^/X 1 Ja-Xc, J~X 9 Ja-x 3 or - tanh - *, (16) V^ la^/Ai since a^, a? 2 , & 3 vanish when p and g are made zero ; and this is equivalent to the result of equation (9), 189, with a= 1/n, A l-a.I-ka and fa Similarly, for the Second Elliptic Integral, 00 = -I* TET f^- C* tanh-"^/-*; 1 --^ (a=x} (17) as before, in 174, 176, and 186. 218. Abel s pseudo-elliptic integrals are derived by making the u s equal in equations (7), (12) ; or the v s equal in equation (13) ; also by making their sum equal to a period a , or the sum of multiples of periods, such s^sp^ + qcoy Now /JL log 0(it, v) is of the form log 2 pu, or <j)(u, vY is of the form e-P M Q, where Q is a rational integral function of <@u and <p u of the form of in (8), sometimes qualified by a divisor ^/(^u e a ). We begin with the simplest case of an algebraical herpolhode by taking < y = ft) 1 + Jw 3 ; and then, from equations (39) and (40), 54, we can infer that the value of s, between e l and e 2 , which makes &JZ3?.iV~ e s _ e i _"~ e 2 e z ~ e z s 8 or v = AND THEIR APPLICATIONS. 229 Denoting pu by s, p u by ^/S, and pv by a, we infer that is pseudo-elliptic, that is, can be expressed in terras of /ds/JS and of i&u-\QJS/P). In fact, by differentiation of _ . since ip v = - 2 /v /(e 1 - e 3 . e 2 - e s ) { ^/(e l - e 3 ) - ^/(^ - e s ) }. In the herpolhode, therefore, of 113, or = - ^ + H s /(e 1 - e s ) - J(e 2 - e.)} nt, and therefore, relatively to axes revolving with constant angular velocity, the herpolhode will be the algebraical curve, given by a s (a s) cos 20 = ^/(s e l .s &,), (a - s) 2 cos 2 2(9 = (a - s) 2 - (e 3 + 2 a )(a - ) + (a - e.Xa - e 2 ), (a - s) 2 sin 2 20 + { VK - 6 3 ) + v/(* 2 - e 3 )} 2 ( a - s) where, as in 1 1 3, a s, or p u pu = 2 7I/" Referred to Cartesian coordinates, in which this equation becomes of the form ( 2 + 6 2 )(i/ 2 +6- 2 ) = a* ....................... (18) 230 ELLIPTIC INTEGRALS IN GENERAL, The relation $v e% = ftt /(e 1 e^.e 2 e s ), combined with the equations of 110, 113, leads to the relation A-D.D-C_A-B.B-C . D 2 B 2 and either B = D, which gives the separating polhode ; or D = A~B + C the relation for this algebraical herpolhode. Now, from 108-110, (D D\ 2 /x 2 (D D\ 2 /x 2 while, with A > B > D > G, and e e = e v e a = e z , e b = e 3 , ^1 ^2 ^c ?a (Q To determine the confocal surface which will describe this algebraical herpolhode by rolling on a fixed tangent plane, we must equate the angular velocity of the axes to jwA 2 /A 2 ; and X 2 1 The squares of the semi-axes of the confocal are therefore _l_ 1 A 2 2 D 1 I B-2- D 1 1D while the square of the distance from the centre of the tangent plane on which this confocal rolls is given by The confocal is therefore a hyperboloid of two sheets, of the 0,2 ,,,2 ~2 -S-tftS- 1 arid in rolling on a fixed tangent plane at a distance b from the centre, it will trace out the algebraical herpolhode (18), being the preceding herpqlhode, changed in scale in the ratio of h to b (Halphen, F. E., II., p. 285). AND THEIR APPLICATIONS. 231 219. A more complicated case can be constructed by taking v a?! + Jo) 3 ; but now we must choose particular numerical values for # 2 and </ 3 . If we select the modular angle of 15, then 2or = J, and in (C), 53, J= 5 3 -7-4, J- 1 = H 2 H-4 ; so that, by choosing A = 108, then # 2 = 15 #3 = 11 ; and i = i-r-V 3 e. 2 =-l, e^ = ^-^3. It is easily verified that, with the above value of v, $>v = ^\ for p2v= f = p4i>; also this value of <pv or s makes, in equa tions (39) and (40), 54, Je, e 9 . e 9 ?o \ if e i~~ e z- e i~ e ^ ^ * ( J -f^ ; fe -) =2 P (--^; h ; ^ -4 The corresponding elliptic integral of the third kind in the herpolhode will now be pseudo-elliptic ; we find, in fact, that, if 0= . sin - 1= (2s -1)* dO 1 2s + 5 1 since i$ v= 3^2 ; so that, in the herpolhode, flip vdu . A <t>-nt=/ - -=-i x /2 ?i^ + 0; y pv-fwi- and therefore, relatively to axes revolving with constant angular velocity /u. ^^/2n, the herpolhode will be the alge braic curve (2s- 1) or (l-2s) 3 sin 2 30 + 9(l- 2s) 2 - 108 = 0, in which 1 - 2s = 2(pv - $>u) = 2^ 2 ^ = 3^, suppose ; 71" /i" C" and now ^sin^O + 3c 2 p 4 - 4c 6 = 0, .................. (19) a curve, consisting of six equal waves, arranged on a circle. With (i.) A > B > D > C, and 2/4 T) T) then (113) so that A-D.D-G A-D.B-D 232 ELLIPTIC INTEGRALS IN GENERAL, Then, either A D = 0, which would give a stable rotation about the axis A ; or zr;B+(7 ; .......................... < 20 > so that D is the harmonic mean between B and C. 3 ^B-D.D-G Again, pv-e a = ^-z- -^c~ ~> so that ~ = which is impossible, with A > B > C. But (ii.), with A > D> B > C, we find that D is the har monic mean between A and B] also i-.I-x/Sf! n 11 2_V3/1_1\ D A~ 2 \D BJ B + C~A~ ~2\C B) so that 2 + ^/3 is the ratio of the semi-axes of the focal ellipse of the momental ellipsoid, and /3(^/3 l) is the excentricity. Another algebraic herpolhode can be constructed by taking |co 3 ; and, with <7 2 = 15, <7 3 = 11, we find that Now, if in . 1 6 ^ y 2(^/3-1) ds~ so that /%p vu_ $>v-pu~ (28-2^/3 + and now the algebraic herpolhode, with respect to revolving axes, is given by (28-2^3 + 5)^1 30 = 6(^/3- I)*J(s-e 2 . s-e 3 \ reducing to an equation of the form ................... (23) AND THEIR APPLICATIONS. 233 With (i.) A > B > D> C, and T , , A-D.D-C -B-D.D-G Therefore -^ -^ and rejecting the factor DC y jD-C .4 2^/3-3 1 1 2,7:i-3/l I - = r ~= - ,, -A-D 0= 6 r so that the excentricity of the focal ellipse of the moniental ellipsoid is ^3 1. With (ii.) A > D > B > C, we are led to an impossible result. Points of Inflexion on the Herpolhodes. 220. The original herpolhodes drawn by Poinsot (Theorie nouvelle de la rotation des corps) were represented with points of inflexion, as curves undulating between two concentric circles on the invariable plane. But it was pointed out by Hess, in 1880, and de Sparre (Comptes Rendus, Nov., 1884), that such points of inflexion can not exist on Poinsot s original herpolhodes, which are curves always concave to the centre, as drawn in Routh s Rigid Dynamics, Chap. IX. ; like the horizontal projection of the path of the bob of a conical pendulum, or like the path of the Moon relative to the Sun, a good figure of which is given in the English Mechanic, p. 337, June, 1891, by Mr. H. P. Slade. The herpolhodes described on planes parallel to the invari able plane in Sylvester s representation are capable, however, of possessing points of inflexion, when the confocal of the momental ellipsoid attains a certain shape. (Hess, Das Rollen einer Fldche zweiten Grades auf einer invariabeln Ebene* Munich, 1880 ; de Sparre, Comptes Rendus, Aug., 1885.) 234 ELLIPTIC INTEGRALS IN GENERAL, Denoting by h the constant distance from the centre of the plane upon which a quadric surface rolls, de Sparre shows that the herpolhode on the plane has points of inflexion, when the quadric is (i.) an ellipsoid 5+S+S- 1 - <<* if W^.o-d^ + i; (in a momental ellipsoid, A<B + C, or z < -i + - i > so ^hat points of inflexion cannot exist on the herpolhode) ; (ii.) a hyperboloid of one sheet /g2 y2 z 2 111 "2 + ?5 ~~ "9 = 1 a 2 <b 2 , if A, 2 < a 2 , and -s > r 2 + ~o a 2 6 2 c 2 a 2 b 2 c 2 (iii.) a hyperboloid of two sheets x 2 v 2 z 2 111 -o T7 , = 1, 1 2 < c 2 , if 70 > -o + -o, whatever the value of /i. a 2 fc a c 2 6 2 a 2 c 2 These herpolhodes being similar to the original herpolhode of the momental ellipsoid, when referred to axes rotating with constant angular velocity /xA 2 /A. 2 , can be considered as defined by the polar coordinates p, 0, given in terms of the time t, by the equations of 113, p fs5S #(fW fW), (1) d9 vifi v /n\ = m + - ~n (2) at $>v $u with u = nt + up v = (0 l + t a) 3 , m/ju = l \ 2 /h 2 . Denoting the velocity in the curve by F, and its radius of curvature by R, then, resolving normally, V*_d P I d( $e\ dO(<]*p R~ dt P ~dt\ p dt) p dt\dt 2 which will be found to reduce to an equation of the form F 3 _ == P / wh ere P = m 3 -f- 3mn 2 $)v + nHp v, Q = $m*nip v mn 2 $>"v ^nH$> "v ; and the corresponding herpolhodes will have points of inflexion when X is chosen so that Pp 2 -f Q can vanish. Thus Halphen points out that the algebraical herpolhode of 218 will have points of inflexion, if b 2 < Ja 2 . AND THEIR APPLICATIONS. 235 221. The polhode being given by the intersection of the two quadric surfaces Ax 2 +% 2 + Cz* =M 2 , we may in consequence write c where (BC)c A(B-C)a* + B(C-A)b 2 and then 2 +X 6 2 +X c 2 +X the equation of a system of confocal quadrics, on choosing I B-CC-AA-B such that I = . 1 D I TT~ Then n n ^A]^ 2 a ~ b - By varying X along the polhode, we find 2 dx 1 d\ dx \ x d\ _ __ ______ _ __ OT" - _ _ _ xdt7a*+\dt ~dt 2o*+X(ft so that the polhode is an orthogonal trajectory of the confocal surfaces, for any one of which X is constant ; and two ellipsoids can be drawn on which the curve is a polhode, of which the generating lines of the confocal hyperboloid through the points are normals. When these confocals are hyperboloids of one sheet, the generating lines may be made of material rods or wires, jointed at the points of crossing ; and now any such a system of rods forming a hyperboloid is capable of deformation, and assumes in succession the shape of the confocal hyperboloids ; the trajectory of any fixed point on a rod being orthogonal to the hyperboloids, and therefore capable of being a polhode, if the hyperboloids are coaxial with the momental ellipsoid of the body. (Messenger of Mathematics, 1878 ; Senate House Solutions for 1878 ; Larmor, Proceedings Cam. Phil. Society, 1884, Jointed Wickerwork] Darboux and Mannheim, Rendus, 1885 and 1886.) 236 ELLIPTIC INTEGRALS IN GENERAL, Darboux has shown (Despeyrous, Cours de mecanique, t. II., Notes XVII., XVIII.) that if we hold a given generator fixed, then any point fixed in any other generator will describe a sphere ; thus, if a rod moves with three points P, Q, R on it connected by means of bars to three fixed centres A, B, G in a straight line, any other point 8 of the rod will describe a sphere about a centre D in the line ABC, such that the A. R. (ABCD) is equal to the A. R (PQRS). The point where the line PQR meets the generator parallel to ABC will describe a plane, the corresponding centre being at an infinite distance ; and generally, if one generator is held fixed, any point on the parallel generator will describe a plane. The herpolhode can now be described by taking a jointed hyperboloid, similar and similarly situated, and of half the size of the former one used for describing the polhode, with one generator fixed along the invariable line 0(7, and with the par allel generator along the normal PQ at P ; and now, if P is moved in a direction perpendicular to the hyperboloid at P, it will describe a plane curve, which is the herpolhode. 222. Any point fixed in a body moving under no forces, whose co-ordinates with respect to the principal axes are represented by a, b, c, will have component velocities cq br, ar cp, bp aq, parallel to the principal axes; and will describe a curve whose projection on the invariable plane will be given, in polar co-ordinates p and 0, by ( 104-113) (bCr- cBqf + (cAp - aCr) 2 + (aBq - bA cff ~~~ + {(a 2 + b 2 )r - cap - bcq }~ > the moment of the velocity about the invariable line OC ; and p, q, r are given as functions of t in 32, IOC, and 108. AND THEIR APPLICATIONS. 237 The equations are much simplified when the point is fixed on one of the principal axes, when two of the three quantities a, b, c vanish ; and it will be a useful exercise for the student to prove that, in these cases, the curve of projection on the invariable plane with respect to axes rotating with angular velocity G/A, G/B, G/C respectively, is given by an equation of the form x + iy = k<f>(u,(a a v), or k<f>(u, w 6 -f), or k(/>(u, (o c -v). Another useful exercise is to deduce -Poinsot s relations when the co-ordinate axes fixed in the body are not principal axes. Now, if the equation of the momental ellipsoid is Ax 2 + By* + Cz 2 - ZA yz - 2B zx - 2C xy = Dh 2 ; and if p, q, r denote as before the component angular velocities, and h lf h. 2 , h s the components of angular momentum about the axes, the three equations of motion under no forces are where h^Ap-C q-B r, h, = Bq-A r-C p, h 3 = Cr-B p-A q; and these equations are solvable by elliptic functions. (Dissertation Ueber die Integration eines Differentialgleich- ung ssy stems ; Paul Hoyer, Berlin, 1879.) 223. The numerical results obtained in the preceding alge braical herpolhodes can be utilized in the corresponding problems of the revolving chain ( 205-206) and of the Tortuous Elastica ( 207). Putting t = J, or v = o> 3 in 206, then p-y = e 3 - J(e^ - e 3 . e. 2 - ej, . 2 - ej} pu-pv ^^^^ or (s-^)cos[2^+ { V(i-*s) + V(2-*3)}W] = ^/(s-e^s-e^ where s-pv = r*/k\ In the corresponding problem of the Tortuous Elastica of 207, it is merely requisite to replace x by the arc s. 238 ELLIPTIC INTEGRALS IN GENERAL, The working out of the analogies for the other algebraical herpolhodes is left as an exercise; merely mentioning that jKK; 15, H)= -f, and that, if = J sin- = j cos- (2s + 3)* (2 dO 1 2s + l 1 ^ * \i<& vdu u I . , W = -75 - , sm - * ^- $m-py ^2 3 (^ 224. The analytical expressions in 208, 210 for the motion of the Spherical Pendulum and of the Top or Gyrostat show, by comparison with the equations of the herpolhode in 200, that this motion may be considered as compounded of two Poinsot representations of the motion of a body under no forces, as given in 104, 214 (Jacobi, Werke, II., p. 477). The relations connecting these two component Poinsot motions have engaged the attention of Darboux (Despeyrous, Cours de me canique, II., Note XIX.), of Halphen (F. E., II., Chap. Ill), and of Routh (Q. J. M., XXIII.). We may put the conclusions arrived at by these mathema ticians in the following condensed form, depending on funda mental dynamical and geometrical considerations. (i.) If the vector OH represents the axis of resultant angular momentum, then H lies in a horizontal plane through the point G, where the vertical vector OG represents G, the constant component of angular momentum about the vertical. (ii.) If the plane drawn through H, perpendicular to the axis of the Top, cuts this axis in G, then 00= Or, the constant com ponent of angular momentum about 00, the axis of the Top. (iii.) These two planes, one horizontal and through G, which we shall call the invariable plane of G, and the other through and perpendicular to OG, which we shall call the invariable plane of G, intersect in a line HK perpendicular to the vertical plane GOG and if HK meets the plane GOO in K, then CR 2 - GH 2 = OK 2 - GK 2 = OG 2 - OG 2 = G 2 - 2 r 2 . (iv.) The instantaneous axis of rotation 01 lies in the plane HOC ; and if 01 meets GH in I, the resultant angular velocity AND THEIR APPLICATIONS. 239 about 01 is OI/C; also CI/CH=C/A, and the velocity of C is r . CI. (v.) By equation (i.) of 210, the square of the velocity of G is (:>C- 2 r 2 Wg/A)(c-hcos6); so that CI 2 = (2C 2 Wg/A)(c - h cos 0), = 2 A Wghk(pw $>u), suppose. Then, by equation (3) of 210, with u = nt + u> %ln-k*p" 2 u = gtf(pu - pa)(pu - &b)(pu - pw) - (a and therefore, when u = a, b, w, we have three equations of the form i$> a = a + /3pa, i$> b = a + /3$>b, if r w so that, according to 165, we may put iv = b a. (vi.) Now GH 2 = 2AWghk{&(b-a)-&i<,} 2 A Wghk(f>w -fru,), suppose, where pi</ - p(a + b) = - (G 2 - C 2 r 2 )/ 2A Wghk ; and since . G-Cr and therefore ptg - f(b - a) = - (p and therefore ( 151) we may put w = (vii.) The point JT moves in the invariable plane of G with velocity equal to the impressed couple of gravity, and parallel to the axis of the couple ; so that the velocity of H is in the direction HK, and equal to Wgh sin ; and the moment of this velocity about G is Wgh sin 6 . GK. But GK sin 6 = OC-OG cos 0, so that p 2 (d<j>/dt) = Wgh(Cr - G cos 6), if p, denote the polar coordinates of H in the invariable plane of G. Now p z = 2 and cos = so that finally we shall find, after reduction, __ _ dt 24 and therefore H describes in the invariable plane of G a her- polhode with parameter 6 + a. 240 ELLIPTIC INTEGRALS IN GENERAL, (viii.) Similar considerations will show that the curve de scribed by H in the invariable plane of C is also a herpolhode, with parameter b a. If in equation (2) of 210 we replace Or by AT, the motion of OC is unaltered, but now the momental ellipsoid at becomes a sphere, and OH is the instantaneous axis of rotation ; so that the motion of OC is produced by rolling the cone, whose base is the herpolhode described by H in the invariable plane of 0, on the cone whose base is the herpolhode in the invariable plane of G, the angular velocity being proportional to OH. (ix.) But in the general case, where 01 is the instantaneous axis, the curve described by / in the invariable plane of G is similar to the curve described by H, and is therefore a herpol hode. Now from (v.), drawing CM, IN perpendicular to OG, 20 2 Wg/A)(c- A -s . GN so that O/ 2 varies as the height of /above a certain horizontal plane ; and the locus of 1 is therefore a sphere, to which the point and this plane are related as limiting point and radical plane. The motion of the Top can therefore be produced by rolling the herpolhode described by / in the invariable plane of G on this sphere, with angular velocity proportional to 01. (x.) It still remains to be shown that the cone described by 01 in space round OG is a herpolhode cone ; this is left as an exercise. Darboux shows that two such hyperboloids as those described in 221, with a pair of generating lines, PQ, PQ in coincidence, and the opposite generators OG, OC of the same system inter secting in a fixed point 0, may be used to represent the motion of OC, the axis of a Top, when OG is held vertical; the point P of intersection of the coincident generators being made to describe herpolhodes in the invariable planes of G and G, by being moved in the direction of the common normal of the hyperboloids. AND THEIR APPLICATIONS. 241 225. The numerical results of the pseudo-elliptic integrals of 218, 219, and 223 can be utilised for the construction oi similar degenerate cases of the motion of the Top. Thus, if a = I o> 3 , b = ^ + io? 3 , then 6 + a = o) 1 + o) 3 , 6 c^ee^; and we shall find cos a = 0, cos ft = K, cZ = sec /3, and C V = 2 A Wgh sec ft, G 2 = 2 A Wgh cos ft. The spherical curve described by C is now given by sin sin(nt cos ft \ts) = ^{cos 0(cos ft cos 6)}, sin cos(nt cos ft \fs) = ^/(l cos ft cos #). With a = Jo> 3 , 6 = M! |o> 3 , and 6 + a = o) v we find that cos a, cos ft, and d are unaltered, but Cr and G are interchanged ; and C now describes the spherical curve sin sin(nt \[r) = *J{cos #(sec/3 cos 0)}, sin cos(nt ^) = ^/(l sec ft cos 0). Again, with a = f o? 3 , 6 = ^ J o> 3 , (/ 2 = 1 5, # 3 = 1 1 : so that ^a= f, p6 = J, we find that and the spherical curve described by G is given by sin 3 $ sin 3\fs = ( 1 2 cos 0)*, sin 3 0cos 3x^ = (l + cos ^ + cos 2 0) v /(2-f 2 cos# cos 2 0). To realise this motion practically, place a homogeneous sphere, of radius c, inside a fixed spherical bowl of radius a, in contact at an angular distance of (50 from the lowest point, and spin the sphere about the common normal with angular velocity The sphere if released will roll on the interior in this curve As another numerical illustration we may take when Also, with 9-2 = 30, p-J3 = - 5 - f /6, G.E.F. Q 242 ELLIPTIC INTEGRALS IN GENERAL, 226. It is convenient to represent the two parts of \/s by ^ and i/r 2 , such that "va-i G + Cr l dt 2 A 1+C080 du 1 G-Cr 1 dt 2 A 1 cos du <pb pu also to put x = V r~V y 2 whence Euler s an^le <p = and dx = Cr-Gcos9 ( cw A sin 2 an expression obtained by interchanging (7 and O in i/r. With ct = pw 3 , 6 = o) 1 + gco 3 , a change of g into q interchanges (} and Cr, while a change of p into > interchanges G and O: both changes of sign change G and G and 6V into Cr, and thus reverse the motion. The following degenerate cases of the motion of the Top will afford an exercise on the preceding results of 210, 224 : A. With b-a = (a lt or q p = Q, ^_ G _ c _ 1 + cos q cos /3 Cr k cos a + cos /3 C 2 r 2 / 2A Wgh = cos a -f cos ft ; and by 215, x is now pseudo-elliptic ; and X ~ \/( cos a ~h cos /3)x/(i#/^X ~~ i , , . / (cos 8 cos 0)(cos cos a) ^J^gJ g /7__foy^-l / \ I / \ / V I + cos a cos /3 (cos a + cos /3)cos i>s/{( cos /3 cos 0)(cos cos a)} = sin " 1 ^- ^ . \ sin t7 _ 1>v /{l + cosa cos /3 ( cos a + cos /3)cos 0|- The angular velocity of H round G in the invariable plane or G is now constant and equal to $G/A. B. With b a = o^ + w 3 , or q p = 1, n_ G _c _I+d cos a Cr~Ji~ cos a + cZ and the spherical curve described by C has cusps on the circle given by 0=8 , and now -, , / (d cos0)(cos0 cos a) where =tan~ \ / - T 7X ^ etc. V 1 + d cos a - (cos a + a)coa AND THEIR APPLICATIONS. 243 The angular velocity of H round G is again equal to %G/A. C. With b + a = u v or q + p = 0, 7 Or 1 -f cos a cos 8. a = -77- = - - - TT > G cos a + cos p and now \Js is pseudo- elliptic, and given by while the angular velocity of H round C in the invariable plane of C is constant and equal to ^ D. With /_> + a = a?! + o> 3 , or g+p = l, (7 r 1 + r? cos a -^= j Cr cos a + a and the angular velocity of # round C in the invariable plane of C is again %Cr/A. E. With g = 1, 6 = o^ + 3 , G Cr = 0, and i/^ 2 disappears ; and now cosj3 = c/A=l, the Top being spun originally in the upright position. Now if the Top falls ultimately to the extreme inclination a, we find that C 2 r 2 /2A Wgh = 1 + cos a ; and subsequently, after a time t, sin ^0 = sin Ja sech[sin \a^/(gjl)t} t Crt /cos cos a . ^2^- 8ln V i so that the integrals for t and i//- are pseudo-elliptic. F. With q = Q ) b = w l ,G Cr = 0, and i/r 2 again disappears; but now cZ = l, and the Top does not rise to the vertical position. For numerical illustrations of this motion, take a = fo> 3 , and </ 2 =lo, # 3 = 11, when ^a=f; or ^r. 2 = 48, ^r 3 = 44, when pa = 4. G. With p = l, a = co 3 , (r + (7r = 0, and ^ disappears; now cos a = 1, and the Top passes through its lowest position. For numerical examples of pseudo-elliptic cases, employ the results pK + ift^; 15, 11) = 1, and ^(^ + -^3; 48, 44) = 2. H. Withp=l and q=I, G = and Cr = ; and the motion reduces to plane revolutions, as in 18. I. With p = l and q = 0, G = and (7r = 0; and the motion reduces to plane oscillations, as in 3. K. With > = !,# =0, d=l, cos/3= 1, cosa= 1, the pen dulum is at rest in its lowest position. 244 ELLIPTIC INTEGRALS IN GENERAL, The Trajectory of a Projectile, for the Cubic Laiv of Re sistance. 227. An immediate application of the function 0(u, v) of 201 occurs in the solution of the motion of a body under gravity in a resisting medium, in which it is assumed that the resistance of the medium is in the direction opposite to motion, and that it varies as the cube of the velocity. Refer the motion to oblique coordinate axes, one Ox in the direction of projection at the point of infinite velocity, and the other Oy drawn vertically downwards. Denote by w the terminal velocity of the projectile in the medium ; so that if W denotes the weight in pounds, the resistance of the air at a velocity v is a force of W(vjw) z pounds, and the retardation produced is g(v/w) B . The equations of motion are then dx = g(ds\ ~ ^AdtJ ds " " ( ) g ds\*dy +J " Eliminating the term due to the resistance, dx d 2 y d 2 x dy _ dx dt dt-~"dP dt~~ 9 !K or, writing p for dy/dx, dp dt dp dx - 01- It Ox makes an angle a with the horizon, then ds 2 _dy* 2 ^^sin</ d&^ dt* 2 di dt** and now equation (1) becomes dfa _ 9( dt 2 ~ uAdt dt AND THEIR APPLICATIONS. 245 Integrating, noticing that dxfdt = oc , when p = 0, - 3 suppose, where _p 3 3^ 2 sin a + 3p is denoted by P; dx _* 2T wP - ^ Then, from (3), d f- = q(~\~ = ^ 9 P > dx *\dt/ ur so that 4^=M w 2 and ^=/p-3 / 7, 1 (6 \ ; (7) while ^ = p, di w n+ f* i (8) 228. The integration required in (6) is similar to that of ex. 8, p. 65, discussed also in 157; we substitute where m is some arbitrary constant factor ; and then 40 s - gr s = { (4m 6 - g 3 )p* - 1 2m*p sin a + 1 2m 6 }/j9 2 , which is a perfect square, when 4m 6 # 3 = 3m 6 sin 2 a , or </ 3 = m 6 (4 3 sinV) ; so that J(z* -flrg) = mV3(2 -p sin a )/p, and dp c/dx on choosing m 2 = J ; so that gx C* & 246 ELLIPTIC INTEGRALS IN GENERAL, Then and supposing x = a at the vertical asymptote, where p= ao , na since ga I ^ = 3 V* = 3 so that ,</a /##_ 2 V~ F i<7 2 ~3p vg cfy 3 w z / lft v or x>= /= = -; (W) dx ,qa ,gx ,ga ,gx" @ ~ - <@ y - ~ o ^/.-2 n,,2 5 /,2 ,, 9 <7tt 6^ 2 r, 5 W* and, integrating, y=f- dx, the equation of the trajectory. It is convenient to write w and v for gx/w 2 and ga/w 2 ; and now to be integrated by the preceding rules of 198. Rationalizing the denominator np r v $ u, it becomes since <7 2 = ; and resolved into linear factors, it becomes where CD, o> 2 denote the imaginary cube roots of unity, viz., = - \ i + ix/ 3 ^ " 2 = - J Now, resolved into partial fractions, | . 2 > v 9 u 2 cov ^ 2 2 "T~ 2-i^ "T" "~W pu 2 $ on making use of the results of 196, when g 2 = Q- Then g y ri PVH/^ ri &v& du+af fift<*>+f d w 2 J 2$v $)u j 2$>(jov $)u J 2$) u?v <pu which is prepared for integration as required in 198; and since AND THEIR APPLICATIONS. 2 pv = log o-(v u)-\- log o-u uv + constant therefore the result of the integration may be expressed by The conditions of Homogeneity of 196 also show that the last equation (13) may be written (TV (TO)! or simply ^ = - 3uv - log ar(v -u)-u> log <r((jov - u) - or subject to the condition that y = 0, when u or x = 0. The equation is left in the complex imaginary form, as there exists no theorem for the expression of ]ogcr(ft>v u) in the form P + iQ; unless we introduce a new function <!>(, a), defined by (Halphen, F. E. t I., p. 151) /"a 229. For the expression of the time t in the trajectory, equation (8) leads to w J pv to-u /I p v+p u 7 r\ <p (0v+$> u, r\ <Q ufa+tiu-j 7;- du+ur/ s ~ -dii+co/ - --du, (lo) 2 tov - to\i J 2 wv -$u J 1 pw*t> - p n when resolved, as before for y, into partial fractions ; so that at y - = - log 0( - u, t<) - o) 2 log 0( - , o>r) - o) log 0( - 16, w 2 v), 01 , , co-log w log ^ s cr % era)? era)- or simply = log a-(v u) o 2 log o-(c*)V u) co log <r(o) 2 y u), (1 6) subject to the condition that = (), when x or = 0. 248 ELLIPTIC INTEGRALS IN GENERAL, By addition, I i - { crV crU j (TV (TCOV (7(JD 2 V (T 3 U and this last term, when expressed in a real form, is equal to (Halphen, F. K, I., p. 232.) This can be proved independently ; for J ~Sv> u 2 $ v <@ u constant 230. For the purpose of the expression of y and t in ascend ing powers of x or u t it is useful to employ the function e v ^ v , which we may denote by i/>-( 1(,, t?) or \fs ; (TV so that \j/( u, v) = a-u (j)( u, v), and i/r = l, when u = (). We may now write gy/w 2 = log i/r( u, v) w log i/r( w, w y) o> 2 log \[s( u, w 2 v), gt/W = log \/s( U,V) ft) 2 log l/r( &, ft)V) ft) log \/r( ^, ft) 2 l ). Differentiating logarithmically, on expanding the second side by Taylor s Theorem ; so that, integrating again, 77 2 /i* 3 77 log^(~u, ?;)=- 2 y^ + ^-^^+..., ............... (18) Then, with </ 2 = 0, and ^a)V = o)^v l etc., / 2 V/ / y-ft)Vi;+..., .......... (19) (20) AND THEIR APPLICATIONS. 249 so that 2 = 3 ^V->i- + ,_ ... (2 1) Oi and here u = gx/ iv 2 , g. 2 = 0, g B = T T T (4 3 sin 2 a), p; = I , 231. When p v p 2 , p 3 denote the values of p corresponding to three points defined by the values x v x. 2 , X B of x, or u v u 2 , u z of u, such that X 1 +x 2 + x 3 = 0, or i6 1 -f it 2 + U B = 0, then, according to 145, This Theorem follows also as a corollary of Abel s Theorem, as applied in 166 ; and it is interesting to proceed to the determination, in a similar manner, of the corresponding values of 2/1 + 2/2 + 2/3 an d *i + ^ + ^ 3 . Changing, in 166, x into p and y into P*, then from (7) 166, n dy 2 + dy 3 )=p l P l - 3 z2 " Therefore = -log(a-l)-colog(a-w)-a) 2 log(a-co 2 ),...(24) = log(a 1 ) - ft) 2 log(a CD) a> log(a co 2 ) ; . . (25) P^_Ps Ps_Pi Ps_Pi where a = - 2 - --=> - JL=J - f_?_ ; ..... (2 6) ^2-^3 Ps -Pi Pi~P 2 and a = ao , when ^ =^> 2 =_p s = 0. As a corollary from the preceding expressions for y and t in terms of x or u, it follows that <r(v u-j)(r(v u 2 )a-(v U B ) _ 1 o- 3 v a-u l <rU^rU B 250 ELLIPTIC INTEGRALS IN GENERAL, 232. By taking x s = and p a = 0, then Pi+PtPiPfina=*O t or l/p 1 + l/p 2 = sin when a3 1 + a; 2 = 0, or Now, from equations (13) and (16), = - log =- J S tan - 2 $> 6 u $rv - 0) = - log + ^3 tan - 2 jp% ^ 3 v) v In particular, when ^^ = a) 2 , then and .^ = _ S 2*= _ 1 log fc^V V3 tan -- ^ 4 fy ^2 V so that the expressions for y and t are pseudo-elliptic ; and, at this point, p = 2sina. 233. We may now investigate the properties of certain points. on the, trajectory. When u = 2o> 2 v, then $>u = J, <gfu = J sin a, and p = cosec a, so that the tangent is perpendicular to Ox. The velocity in the trajectory is given by iv(p* - 2p sin a + !)*( p 3 - 3^> 2 sin a + 3|?)~*, and this is a minimum, by logarithmic differentiation, when j9 sing p 2 2p sing + I_ _ 2 2 si n g -f 1 5 3 ~ or p 2 cos 2 a +^ sin g- 1 = ................... (27) If the tangent AB makes an angle ft with Ox at the point A, ^ sin 8 then p=. f , C0s(g p) so that the relation becomes tang=-2cot2/3 = tan/3-cot/3 .......... (28) Then ^(4 + tan 2 g) = tan /3 + cot /3 = 2 cosec 2/3, or (3#) = JC 4 ~ 3 sin 2 g) - cos g cosec 28. AND THEIR APPLICATIONS. 251 The relation (28) is equivalent to a number of other re lations, such as tan(2/3 a) = tan a tan 2/3 = tan a + 2 cot a, tan a ={cot(a-)}*-{ten(a -)}*, etc. Also, since p = -: r~r~* sin a o#> u therefore, at these points of minimum velocity, ^ / 2 w , = T.(4_3sm 2 a) = 3// 3 , and p 3 w=r/ 3 , and therefore ?2?<, = #w, or u = fo> 2 , as in 160. The integrals for y and t at these points of minimum velocity are therefore pseudo-elliptic, and depend on f ds , f sds J (**- iM 4 * 8 - 1) a ./ (s 8 - we** 8 - iy integrals first considered by Euler (Legendre, F. E., I., Cha|>. XXVI). We find, by differentiation, that (29) " , ...(30) ^ts- i; tj ^-tr L) -f x/ o V ^" i ; -^tan- 1 x/(4* 3 -l) (2 by means of which the results can be constructed : and noticing that, if s = <pv, *J(4-s 3 1 ) = p v, <j<, = 0, g z = 1. then we find finally, when u = |cD 2 , n-Vly-f^), (32) 252 ELLIPTIC INTEGRALS IN GENERAL, 234. Denoting by the angle which the tangent at any point makes with Ox, the tangent at 0, the point of infinite velocity, and by the angle which it makes with the tangent at A, the point of minimum velocity, then = /3 ( j) 1 and sin sin(/3 <) L) ~ cos( a - 0) ~~ cos(a cosq so that and and since sn ~ 0) - 2 cos ( - ft + 0) tan siD - c< s(/3 - 0) - cot 25 sin(/3 - A) - - - -- = - 2 cos a _ _9 = z cos a cosec zp -r-^ - sm ~ 3 sin 2 a) = | cos a cosec ,, r , therefore . ^~^= \-^ -, = -^... ...(34) tan p <pu-\-$ co 2 Therefore, at points defined by u v u 2 , where the tangents make equal angles with the tangent at A, p tt 1 *P /t i ss f *i< k >r Thus, if 1^ = 0, then u^ = w. 2 \ and the tangent where u = u>. 2 makes an angle 2/3 with Ox. By the principle of Homogeneity of 196, we can select any arbitrary value of </ 3 , and it is convenient to take # 3 =1; and gx u , l/ -^ = , then K; 2 m .,. now, if where m 6 =g 3 , With ,(/ 2 = 0, # 3 = 1, we have found, in 166, . v (///. ,, Again, n - =^, then m w 2 <pv = (4 - 3 sm 2 a)-*, ^ = ^/3 sin a(4 - so that, as a increases from to JTT, and v increases from w. 2 to |o) 2 . cos v increases from to AND THEIR APPLICATIONS. 253 Denoting the analytical expression for tan^/tan/3 in (34) by X, then X is independent of a or /3, and therefore a Table of numerical values of X, with u or mgxju^ for argument, will serve for all trajectories. It will be a useful numerical exercise for the student to prove that corresponding values of u and X are 2J/-2 , 1 21 0; i2 |0) 2 , 00 . EXAMPLES. Prove that, with g. 2 = Q, # 3 = 1, 2. ^ (it - f ft? 9 ) = 3. u- /* ^-^ / > j. \ i ,2cm 1 . / _ = f u f .73 tanh - */3 ypit-1 9V p % ^ " T ^" log p ^ w " ^2) iW 3 tan " A log F(U ~ *^) + A v/3 tan 7. Integrate CHAPTER VIII. THE DOUBLE PERIODICITY OF THE ELLIPTIC FUNCTIONS. 235. Besides pointing out the advantage of the direct Ellip tic Functions obtained by the inversion of the Elliptic Integrals ( 5), Abel made an equally important step (Crelle, II., 1827) in showing that the Elliptic Functions are doubly -periodic functions, having a real period, 4>K or 2K, as already defined in 11, and an imaginary period, 4>K i or ZK i, where, as before in 11, o Doubly-periodic functions make their appearance when we consider functions of a complex argument w = u + vi. Denoting x + yi by z y we have already discussed in 179 the system of confocal conies given by rj = c sin w, or c cos w, when u or v is constant. T ,,. r dz In this case w= I , 2 2 -, J v ( c ^ ) and the poles of this integral, as defined in 54, are given by z = c, the foci of the confocal system of conies. Changing the origin to a focus, then r dz iv =1 j- and z = 2c sin 2 \w, 2c z = 2c cos 2 Jt0, dz/dw = c sin w. Denoting by r, r the focal distances of a point, then r 2 = (x 4- yi)(x yi) = 4c 2 sin 2 J(u 4- m)sin 2 J(i, w), 254 DOUBLE PERIODICITY OF ELLIFIIC FUNCTIONS. 255 or r= 2c sin ? = 2c cos so that / + r = 2c cos vi = 2c cosh i;, / r= 2c cosu, giving the contbcal ellipses and hyperbolas, for which v and u are constants. It is convenient to denote x yi by z and u vi by i(/ ; and now the Jacobian T 3(fl5, ?/) 9 . . , J or -) = c 2 sm w sm ^o = Jrf . 9(tt, v) 236. Now, if we consider the integral (11) of 38, then z = sn 2 ^ \z cn dz/diu = sn \w en \w dn |i(; ; and the poles of the integral are given by z = Q, 1, and 1/&. Denoting by r, r , r" the distances of a point from these poles or foci 0, (7, 0" in fig. 26, then r = sn Jitf sn Jt(/, r = en Jw en 10 , &r" = dn \w dn |t(/ ; or by means of formulas (2), (3), (5), (28), (29) of 137, with \w and Ji{/for it and t , and therefore u and i?j for u + v and it v, cnri cnu 1 dn-yi dn u r = K- _ u 4- en u dn t i /c /2 dnvi dnu /cr" = ^^^ en vi dn 16 en u dn ri cnu cnvidntt cnuckivi From these relations, by the alternate elimination of u and r, r + r dn m = en vi\ r rdnu =wu ) or kr" + ^7^ cn i?i = dn vi"\ kr" kr cn u = dn u ) or ki-"du vi kr en vi = 1 k\ kr"&i\ u kr cnu =1 k) the vectorial equations of one and the same system of confocal orthogonal Cartesian Ovals (fig. 26) ; also J=krrr". (Darboux, Annales scientifiques de Vecole normale supe rieure, IV., 1867.) 256 THE DOUBLE PERIODICITY As we travel round one of these curves and make complete circuits, each enclosing a pair of poles of the integral w, defined either by and 1, or 1 and l/k, the integral increases by constant quantities 4K" or 4<K i, the corresponding periods of the elliptic function sn 2 Jt(;, as in 55. y Fig. 26. By making k = 0, we obtain the degenerate case of the confocal conies, and now K=^TT, while K =x> , so that the circular functions have a real period 2-rr and an infinite imaginary period; on the other hand, the hyperbolic functions, as illustrated by the confocal ellipses, have an infinite real period and an imaginary period 2?ri. Mr. J. Hammond has shown, in the American Journal of Mathematics, vol. I., how these Cartesian Ovals may be de scribed mechanically, by means of reels of thread, as in the case of the confocal conies of 173. He takes two reels of thread, of different diameters, fastened together, and pivoted on the same axis at C. Now, if the threads are led through a pair of the foci, and , the curves r IT = c will be described, if the diameters are in the ratio of I to 1. By leading the threads round an oval, as in fig. 26, theorems can be obtained, connecting arcs of confocal Cartesian Ovals, analogous to those of Graves and Chasles for elliptic arcs. OF THE ELLIPTIC FUNCTIONS. 257 237. By inversion of this system of confocal Cartesian Ovals, we shall obtain another system of orthogonal quartic curves, with four coney clic foci A, B, C, D, defined by the vectors z = a, ft, y, S, suppose ; and now w=fdz/J(za . z-/3 . z- 7 . z-S) ; or, writing iv for iv/^/(a y fi~ <>)> then, from 66, 6 S.z a 91 a 8.z 8 01 a 8.z y , 91 - 7) = sn 2 ^(;, ? 75 ==cn 2 iic?, - J = dn 2 to. a-S.z-/3 2 a-<5.0-/3 a-y.z-/3 Denoting by r v r 2 , r 3 , r 4 the distances of a point from the foci A, B, C, D, then, from these equations, Q $ rp R T mod. " . -i = sn ^iv sn Aiv , mod. - = en to en Aw 7 , a o T 2 a o r z mod. - = dn &w dn Ati; ; a-y r 2 so that we obtain the vectorial equations of these orthogonal quartic curves on replacing r t r, r" in the equations of the Cartesian Ovals by these expressions. (Proc. Cam. Phil. Society, vol. IV. ; Holzmuller, Einfuhrung in die Theorie der isogonalen Vei^wandtschaften, 1882.) 238. We now proceed to express the elliptic functions of the imaginary argument vi by functions of a real argument v. We know that cos vi = cosh v, sin vi = i sinh v, tan vi = i tanh v ; and that the function <p or amh u, and its inverse function u or amh~ 1 = log(sec ^> + tan 0)= cosh" ^ec^, etc., connects the circular functions of <f>, for which /c = 0, with the hyperbolic functions of u in 16, for which K = 1 ; and then cosh u = sec 0, sinh u tan 0, tanh u = sin 0, tanh Ju = tan J^>, Now, if (p = amh i/ri, then cos ^ cosh i/ri = 1 , or cos0 cos *// = 1, a symmetrical relation, so that ^ = amh 0/i ; and sin (p = tanh \fsi = i tan ^, cos (p = sech i//>i = sec \^, tan 0= sinh \fsi = i sin i/r, etc. Also d<p = i sech \lsid\fs = i sec A(0, ) = s/(! + ^ 2 tan 2 ^) = sec so that G.E.F. 258 THE DOUBLE PERIODICITY If \/s = a.m(v, K ), then = am(w, /c); and sn(vi, K) = i *,., or isc(v, K), or HU(V,K); cn(vi, K) = TTXI or nc(v, * ); cu(v, K ) , . . . du(v, K) , , dn(m,*) = ^Yy or dc(v,*). connecting the elliptic functions of imaginary argument vi and modulus /c with the elliptic functions of real argument v and complementary modulus //. Putting v = K , we notice that sn K i, en K i, and dn TTi are infinite; and putting v = 2K , then also sn4 J ST / ^ = 0, en ^K i= 1, dn4^ i= 1. 239. The Addition Theorems of 116 may now be written cn(u+ vi*) = (cuu cnv isn udnusn sn(u + vi) (snudiiv -\-icnudny. snv dn(u + vi) = (dn u en v dn -y ?/c 2 sn u en 16 sn v) -=- .D, jD = en 2 v + /c 2 sn 2 u sn 2 v ; remembering that the modulus of the elliptic functions of v is K, while that of the functions of u is K. Thus, putting v K t 1 ,., .cnu = , dn(u+JTi)=-t --- : /csnu snu so that, putting u = K, w(K+ K i) = - IK IK, sn(K + K i) = I/K, dn(K + K i) = 0. Writing C, S, D for en 2u, sn 2u, dn 2%, then ( 123) Generally, when m and TI denote any integers, we find that cn(t6 + 2mK+ 2nK i) = ( - 1 ) m+n cn u, K i) = (-l) m sn u, 2nK i) = (-l) n duu- so that 4fK and ZK i are the periods of sn u, 2K and 4*K i are the periods of dn u ; the periods of cu u being 2(K+K i) and 2(KK i). OF THE ELLIPTIC FUNCTIONS. 259 In 164, we may now write MI + ^2 + u s = 4wiK + 4<nK i ; or in the notation of the Theory of Numbers, ^ + ^2 + ^3 = (mod. 4>K, K i\ 240. A combination of the transformations of 29 and 238, to the reciprocal and to the complementary modulus, gives , . x 1 1 cnfc w, iic/ic) cn(m, K)= 7 - K= -3-7-7 v-/-/\= i / / , /( cn(v, K ) dn(r v, I/K ) dn(/c w, tc/* ) . ^_ ^~ K dn( K vi, Thus cn^ w, i/c//cO = cd(w,, /c) = sn(E w,, AC), or am(/c u, //c/V) = i?r am(^T u, K) ; as is otherwise evident, when we notice that, if u =f (1 - * 2 cos 2 0)~*cty = ^/ (1 + ^ sinV) so that ^ = am(/c / u, i/c/V), then K-u=f(l- K -cos -\!s)~*d\!s= f (l-/c 2 sin 2 0)"^0, ^ o or (j) = a,m(K U,K), provided i/r = JTT 0. 241. As an application, take the values of v 1 and v 2 in 210 ; l + cos/3 cZ cos a cZ + 1 2/ 2 , v , 1 , _ 1 1+cosa 1 + cosa 1 + cos a 1 cos 8 o cZ cos a o cZ 1 . dnV 7 = 1 - --, sn 2 ^ 9 = ^ , cn-?; 9 = _- > 1 cos a 1 cos a 1 cos a so that, with v l =pK i ) v. 2 = K+qK i, where p and q are real proper fractions ( 56), then 1 cos a _ _ snH j _ sn~pl i dn z qK i 1+cosa .sn 2 v cn 2 l i 1 cos/3 KU Z V I dn 2/ y 2 _ K^stfpK i f+cos 8 == ~.sn 2 y dn = " d + 200 THE DOUBLE PERIODICITY Thence, expressed in a real form, 1-cosa = 1 + cos a ~ or ( 135) tanJa- a= Also ( 29) so that = am{(2>+gX# , I//} -fam{(p -)/# , I//}. And or = In the Spherical Pendulum, O = ; and therefore ( 210) 1 cos a 1 cos /3 d 1 _ -. t I+cosa l + cos/3 cZ + "l~ , c^ 1 ancl /r/ -. z^/ en g/^ dn g^ or sn(p g)^T = sn pK cn qK du qK . Thence 8n(g+y)JT cn(q+p)K " <)S/ ^- " ( " 242. With Jacobi s notation of 189, the expression for i in 210 becomes -. - \ sn en -+ ZV,+ 1 snv and now, if we divide i/r into its secular and periodic part, i n the form \/r = ^fu/K + \/r , then ^ is called the apsidal angle, in the motion of the Top or of the Spherical Pendulum, as seen illustrated for instance in a Giant Stride and 2 which must now be expressed in a real form. OF THE ELLIPTIC FUNCTIONS. 261 From 172, rvi iZ(vi, K ) = i/(dn 2 vi-E/K)dvi E fdn*(v, K) i = ^v/ - 9 ; , dv K J cv 2 (v, K ) E . t snvdnv , -K. en v sn v dn v 7 ^ + en v sn v dn i? by means of Legendre s relation of 171. Thus, with v l =pK i, Again, by (2)*, 186, since Z#=0, Z(^+ M.) = Zu - therefore, with v 2 = Also, if /> and ^ are proper fractions, the logarithmic term of i vanishes ( 264) ; so that, finally, In the Spherical Pendulum, n pK /sn pK = /c 2 sn pZ*sn qK si\(p q)K so that = With the Weierstrass notation, taking u in equation (8) of 208 between the limits o> 3 and ^ + 0)3, we find ( 278) i = (a where a = In small oscillations near the lowest position, p and K are very nearly unity, while q and K are small. 262 THE DOUBLE PERIODICITY The Geometry of the Cartesian Oval. 243. Denote the angles POO , POO, P0"0 in fig. 26 by 0, 9 , 6" respectively ; then with as origin, _cu^w cn^w _ /A en n 1 cnw\ ~~cn w+cn w ~ \ \l 1 + cnw/ or, in a real form, with modulus K for the functions of v, ... l/I ~ \ \1 cnu 1 cni>\ snjudnju en u en ~ . n cos ^ = T , sin 9 = , . + en u en v 1 + en u en v With /x as origin, and, similarly, ^,_ ~ _ //] ~ \\l . ^,_ dn \w _ l/l duu 1 dn m\ ~dnw + dnii/"V\l + dnu l+dnvi) sn M . n// cos = -T - , sin 0" = dn Ju /c 2 snusnv dni + dnucnt/ dn ^ + dn ucn v With as origin, and a; -j- yi = sn 2 Jty, ,, i A/ 7 then ^ tan W = sn To reduce this to a real form, similar to the above, we require two new formulas, not included in Jacobi s list ( 137), but easily derivable from it, namely, Now, with \w and \iv for u and v t and u and vi for and u v, dnu + cnu dnm // A/(j \\dnu-cnu dn ,_ //dnu+cnu 1 dn v\ _ en |u dn \u sn |v en |t; ~ V. Vdnu-cnu 1 + dn v/ " sn Jw dn Jv /c /2 sn u sn v cos 6> = 7- i sm (7 = , dnu cnudnv dnu cuuduv OF THE ELLIPTIC FUNCTIONS. 263 244. Again, denoting the angles which P subtends at O O", 0"0, 00 by 0, , <j>" respectively, so that = 7r -<9 -0", $ = 0-6", 0" = 7r-0-0 ; then we shall find tan i,_ sn i^ dn ^ cn i^ //I cnu l+cnv\ en tt sn Jv dn Jt . . /c sn Jit /c sn ^cn ^ for* i (4, 5 S ^ > Vu+cnit 1-cnv/ rV l/dnu cnu 1 dni>\ cn ju dn ^it dn %v . sn i& cn ^u cn ^v dn &v V \dnu + cnu 1+dnv/ 1 dn Ju sn i ^ "A Vl + dnit dnv cn W cn u cn v . sn it sn v cos = ,- , sin d> = - , 1 cnucnv 1 cnucnv , cn w + dn u dn v . , ic /2 sn u sn v cos0=-r ^ , sm0=- r - = . dn u + cn u an v dn it + cn u dn v cn v + dn u dn v /c 2 sn u sn v cos0 = j , sm0 =-= . dn t; dn u cn tr dn v dn % cn v Similarly, denoting by o>, a/, a/ the angles which the normal at P to the oval along which v is constant makes with PO, PO , PO", we shall find , sn u cn v sn it dn v . sn u tan o> = - , tan to = -5 , tan i . UCVXJL \JJ dn u sn v cn u sn v Drawing the three circles through O PO", 0"PO, OPO , and denoting the points in which the normal at P meets them again by Q, Q , Q", we shall obtain similar simple expressions for PQ, OQ, ... (Williamson, Liff. and Int. Calculus). 245. The two ovals defined by v and IK v form a complete curve; and so also the ovals defined by u and 2KU. Denoting by P, P , Q, Q the four corresponding points defined by (u, v), (u, 2K -v), (2K-u, v\ (2K-u, ZK -v)\ and denoting by p, p, q, q their consecutive positions when u receives a small increment du, then Pp = tJJdu = K^/(rr r")du _ cn vi dn u + cn u dn vi //cn vi cn u\j dn vi + dn u \ Vcn m -f cn u) _dnu-fcnudnt> //I cn u cn v\ , dn ^ + dn it cn v \ \1 + cn u cn v/ and changing u into 2Ku, v into %K v, Q, f _ dn M- cn u dn w //I cn u cn t\ , dn v dn it cn t \ \1 + cn t/ cn v/ 264 THE DOUBLE PERIODICITY TU 73^ i r\> > 2 dn^dnv //I cnucnt>\ , Inen rp + tyq =^ A/IT Jdu tc 1 + en w CD v y VI + en u en v/ -y - 2 ^(1 2 so that the sum of the arcs described by P and Q is expressible as an elliptic arc. D r\> > /c /2 cn v , Again Pp Qq=^ -= ~\(T~, -cut, /c 2 1 4- en u en i> \ \l-f-cni6 en t> which is expressible in the form - 2 dn v cos 2 "*" 2. 2 >v/( dn2 ^ + 2 en V dn v cos 0" + cn so that the difference of the arcs described by P and Q is expressible by the sum of two elliptic arcs ; and thus the arc of the Cartesian Oval described by P is given by means of three elliptic arcs, which is Genocchi s Theorem (Annali di Matematica, VI., 1864 ; Mr. S. Roberts, Proc. L. M. S., III., V.). 246. Let us examine the analytical properties and physical applications of the functions log en Jw, log sn ^w, log dn \w. Denoting log en \w by fa + ityv when resolved into its real and imaginary part, then = | log en %w en \w -f | log en J , en \w dn \w en ^w dn \w . _ , .en w r en = 9 lOff - ;; - - ^ - - - f - -- \~ 1 tan 1 ;; - - ^ - - - f - -- ~ - - 7 - j - dn Jto dn \w en \w + en %w en iv dnu + dnmcnu ., ,. Il cnu 1 cnm as in 236, by means of formulas (3), (20), (28) of 137 ; and now expressing the elliptic functions of vi t to modulus /c, in terms of functions of v, to modulus /c understood ; then _il ^dnu + cnudnt; _ 1 l/lcnu 1 en tA ^ l ~ to dn v + dnucn f ^~ \\l + cni6 1 Denoting logsn \w by (f> 2 + i\ls 2 , then = J log sn Jwsn Jty + J logsn J , sn Aw dn iit; sn A^ dn itt; . . , .sn \w sn log ^ -4-^ - hitan-H f y dn %W dn J^ sn \w + sn cnm en u , .. ,. //dnu-fcntt dnvi cnvi ~ ,. //dnu-fcntt dnvi cnvi\ ~H A /(-i --- -r ^ :) \\dnu en u dnvi-fcntrt-/ _j, ~ OF THE ELLIPTIC FUNCTIONS. 265 1 cnucn-y 2 cn . _ 1 //dnu + cnu 1 dnv\ v \\dnu-cnu 1-fdn-y/ Similar!} 7 , denoting logdn^iv by fa + i fiy it cuviduu + cuuduvi . .. //I -dim l- ~ : A . en w + en u \\l + dnu 1-fdnw/ idnu dnv c l/ 1 */! \ \ 1+cnucnt; \\l-r-dni6 By (20), (21), (22), (23) of 137, we prove, in a similar manner, l + cn-it . cuvi-\-cuu , , A _, isuvidiiu 1 emu 2 cnvi en = tanh -1 (cn u en = tanh ~ J (dn u en v/dn v} i tan ~ 1 (cn u sn v/sn u), y ) = etc. i w en w 247. These conjugate functions ^ and -^ of the complex u-\-vi are capable of representing the solution of various physi cal problems concerning a plane in which u and v are taken as rectangular co-ordinates, since they satisfy the conditions 3u ~dv *dv 3i6 Here tt and v are not restricted to be rectangular co-ordinates, but they may represent the conjugate functions of confocal conies or Cartesian Ovals, as in 179, 236, or of any orthogonal system, which divides up a plane into elementary squares or rectangles, as on a map or chart. As in 54?, we take a period rectangle OABC, bounded by u = 0, u=2K, v = y v = 27i ; and now, as the end of the vector w or u + vi, drawn from 0, travels round the boundary OABC of this period rectangle, the vector w assumes the values ZtK(Q <*<!); 2#+ 2* / JGT i(0 <$ <!); 2tK+2K i(I > t > 0) ; 2 JTi(l >t > 0). When the sides of the period rectangle are a and 6, we replace u and v by 2Kx/a and 2K y/b, where K /K=b/a. 266 THE DOUBLE PERIODICITY Taking the function log en \w or ^-}-i^ v then from to A, V^i = ; from A to B, ^ = ITT ; from to (7, ^1=2^; an d fr m G to 0, ^ = 0. At A, where K = 2^T, -y = 0, then 1= = oo ; and at (7, where u = 0, -y = 2K, ^ = oo . The functions ^ and i/r lf therefore satisfy the conditions required of the potential and stream function, due to electrodes at A and C, of the plane motion of electricity or fluid, when bounded by the rectangle OABC. The function ^ will also represent the stationary tempera ture at any point of the rectangle, when the sides OA, OC are maintained at temperature zero, and the sides AB, BG at temperature JTT. When the period rectangle is a square, or KK\ then \^i- i 71 " when u+v = 2K, or along the diagonal AC; we thus obtain the permanent temperature inside an isosceles rect angular prism, when the base is maintained at one constant temperature, and the sides at another. Similar considerations will show that the function logsnjw or 02 + ^2 w iH &i ye ^ e streaming motion in the same period rectangle, due to a source at 0, and an equal sink at C. The function \js z is now zero along OA, AB, EG, and JTT along 0(7; and ^ 2 will therefore represent the stationary temperature when OC is maintained at temperature JTT, while the other sides are maintained at zero temperature. A superposition of four such cases will give the permanent temperature when the sides of the period rectangle are main tained at any four arbitrary constant temperatures. (F. Purser, Messenger of Mathematics, VI., p. 137.) EXAMPLES. 1. Solve the equation 2. Investigate the curves given by dz/dw = (I-z*)$. 3. Prove that the system of orthogonal curves given by are the stereographic projections of a system of confocal sphero- conics ( W. Burnside, Messenger of Mathematics, XX.). OF THE ELLIPTIC FUNCTIONS. 267 Prove that the stereographic projection of the points on the sphere whose latitude and longitude are 6, <p, are given by Prove also that ^ toy f%y /as y _ /^y (*y\* n* va) + v^J ^ \d~J ~\^v) + w + W 4. Discuss the physical interpretation of -ic/c sn u sn v . .ic cnv . dn u dn -y /c en u and determine the single function from which it is derived ; K sn u sn v f , . . . , also of 0-h^0 = tanh~ 1 J +^tan J dn n dn v en v Interpret these expressions when 5. Prove that, if x + yi = sn w, then gives the plane motion of liquid streaming past two obstacles given by x = l and l//c, x= 1 and l/ K (W. Burnside, Messenger, XX.). 27ie Double Periodicity of Weierstrass s Functions. 248. A procedure similar to that of 236 will show that the Cartesian Ovals of fig. 26 are also the representation of the conjugate functions of the system z ^w, obtained from the definition of 50, r dz or dz/dw = p w=- where 4s 3 -g^-g 3 = 4(2 - e^(z - e. 2 }(z - e a ) ; and z = e lt e 2 , e B define the three foci. According to 51, &iv -e z = (e l - 6 3 )ds V(^i ~ ^)w = (e 2 - e 8 ) cn pw - 6l = (e x - e 3 ) cs V^ - e^tv =-(e l - ^ 3 )dn by 239 ; thus identifying these results with those of 236. 268 THE DOUBLE PERIODICITY With the notation of 202, and denoting the focal distances by r lt r 2 , r 3 , and u vi by w , 249. To express these focal distances in a real form, as in 236, we employ the Addition Theorem (K) of 200, written a-(u + v)v(u v) = o-% cr 2 f { (pv e a ) (<pu e a ) } = (r 2 uo- a -v-cr a 2 u ( r 2 v .................... (M) Again, from 154, $>(u + v) e a is a perfect square; and we may write x = pu t y = $>v, s = p(u+v) > _ -eg.pv-ep.pv- e y ) - */(pu-ep . pu-e y . $>v-e a ) N $>v $>u and now <r a (w + v)a-(u v) = +J { p(u + v) Co] cru ar 2 v (<pv <pu) = VU <T a U (TpV (T y V <TpU <T y U (T^V (TV,... (0) and changing the sign of v, ar(u + v)a- a (u v) = o-U (r a u oyy o- y v + cr^U o- y u a- a v crv. . . .(P) Again, by multiplication with (N) and reduction, v) or <r a ( u + v )<rp( u ~ v) = ar a u a-pU a- a v a^v - (e a - e^a-u a- y u <rv <r y v, (Q) cr a (u v)arp(u + v) = o- a u a-pU a- a v a-pV -f (e a -e^)a-u a y u crv cr y v. (R) Similarly, (u -v) = (pu- e a )(^v - e a ) - (e a - e ft )(e a - e y ) cr(u--v) <pv <pu or <7 a (u + v)o- a (u - v) = (r a ~u cr a 2 v - (e a - 6p)(e a - e y )ar 2 u <r*v ....... (8) (Schwarz, Elliptische Functionen, p. 51.) OF THE ELLIPTIC FUNCTIONS. 269 Now, from these equations (0), (P), (Q), (R), with w or %(u+vi) for u, and w or ^(u vi) for v, _ a-w a-iv <r<tf(r.v , vii (rvi or r = with similar equations for r 2 and r 3 ; and thence the vectorial equations of the Cartesian Ovals analogous to those of 236 r z <r s u - r B a- 2 u = (e. 2 - e^u \ etc ^ r 2 o- 3 t i r 3 <r. 2 v i = (e 2 ejvjvij These vectorial equations again are the geometrical inter pretation of the formula, immediately deducible from (N), (T) Making m 2 = 1 in the homogeneity equations of 196, gives V( 5 92 &) = ~ V( v 5 02> -0s) the equivalent of the equations of 238, by which a change is made to a real argument and complementary modulus ; while (w; 02> 3 )= - 4(^5 02 -9s) -(vi , 2 . 3 )= ^>; 2 . -0 3 ) o- a (^; 2 3 )= <r a ( v > 02 -0s)- 250. When a point has made a complete circuit of one of the ovals, enclosing a pair of foci, defined by e. 2 and e 3 , or e l and e 2 , z will have regained its original value, but w will have increased or diminished by 2^ or 2o> 3 , defined as in 51, 52 by the rectilinear integrals so that 2^!, 2o) 3 are the periods of the function pw, and To fix the ideas we have supposed the circuit of two poles of the integral made on the enclosing branch of a Cartesian Oval, but the result will be the same whatever be the curve, provided it makes the same number and nature of circuits. Now, in 165, we can have EE() (mod.20)!, 2o> 3 ). 270 THE DOUBLE PERIODICITY 251. In 54 it has been shown how, as the vector of the argument w traces out the contour of the period rectangle, <pw assumes all real values : and $w may be made to assume any arbitrary complex value at a point in the interior of the rectangle, given by a determinate vector t^ + t w B . It is convenient to put o) 1 + a) 3 = o> 2 , so that ft>! + w 2 + w 3 = 0, with gj_ + 6 2 + e s ; and now ^o^ = e v $a) 2 = e 2 , #>o> 3 = e 3 ; while %> u>i ~ fi coz = $> to 3 = 0. The equations of 54 show that e-. e 9 . 6, Co A equations analogous to those of 57, in Jacobi s notation. Thus, from ex. 9, p. 174, With negative discriminant, as in 62, we take e 2 as real, and e v e 2 imaginary; also o^ = J(w 2 + a/ 2 ), o> 3 = J( W 2 ~~ w/ 2) > 252. A great advantage of the Weierstrassian notation (at first rather baffling to one accustomed to the methods of Legendre and Jacob!) is that the dimensions of the elliptic integral are left arbitrary, and can be changed by an applica tion of the Principle, of Homogeneity of 196. When the canonical elliptic integral of 50 is normalized in Klein s manner ( 196) by multiplying by A T \ then A^eZs r da- where s = AV, (/ 2 = A^y 2 , g 3 = and now y> 3 27y 3 2 = 1, so that the new discriminant is unity, and T O T" 1 \*w O If GT P trr 3 denote the real and imaginary half periods of the normalized integral, then = ft) 3 OF THE ELLIPTIC FUNCTIONS. 971 The general elliptic integral, written with homogeneous variables as in 155, is also normalized by Klein by multiply ing by the twelfth root of the discriminant of the corresponding quartic, and its half periods are now c^ and C7 3 . If we normalize, for instance, the canonical integral (11) of 38, written with homogeneous variables x v X 2 , in the form f(x^x. 2 .x 2 x l .x. 2 kxjrtyfyfa^ x l dx. 2 ) > then the invariants g z , g 3 , and the discriminant A of the quartic 12 * 2 ^~ 1 " *^"> ~ A/tX/-fl y being the expressions given in 68, therefore Now the half periods of integral (11), 38, being 2K, 2K i, We are thereby enabled to change from Weierstrass s (a l and o> 3 to Jacobi s K and K, and to utilize the numerical results of Legendre s Tables. (Klein, Math. Ann., XIV., p. 118.) When the discriminant A is negative, we normalize by multiplying by ( A) 1 ^, and replace o^ and o) 3 by u> 2 and o>./ ( 62); but now the new discriminant y 9 3 27y 3 2 = 1, and o>. 2 ( - A)" = SA^/Q/o/), o/,( - A^) = 2K i 4/(^ ) ( 47, 58). For instance, if g 2 = in 50, (-A)^=^/3^/(/ 8 ; and in 58, ,7=0, or 2/C/-J, 24/(J^ ) = 4/2; and now while ( 47) wjVwj = K ilK= i Confocal Quadric Surfaces. 253. The symmetry and elegance of the Weierstrass notation is well exhibited in the physical applications relating to con- focal surfaces of the second degree. The equation of any one of a system of confocal quadrics we put a 2 + \= and now the interal d\ X With e l >e. 2 > e 3 , we must take a- <b 2 < c 2 . 272 THE DOUBLE PERIODICITY Three confocals can be drawn through any point x, y, z, an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. Supposing the ellipsoid to be defined by X or u, and the hyperboloid of one sheet in a similar manner by fj. or v, and the hyperboloid of two sheets by v or w ; then in going round the period rectangle of 54, (i.) u =pco, oo > pu > e v for the ellipsoids ; starting with p = for the infinite sphere, and ending with p = \ for the inside of focal ellipse; (ii.) v = coj + <?ft>3, e l >pv>e 2 , for the hyperboloids of one sheet; starting with q = from the focal ellipse, and ending with (7 = 1 for the focal hyperbola; (iii.) W = ?*! + o> 3 , e 2 >$w>e s , for the hyperboloids of two sheets ; starting with q = l. from the focal hyperbola, and ending with q = for the outside of the focal ellipse ; (iv.) the fourth side of the period rectangle gives imaginary surfaces. 254. Replacing 6 2 -a 2 and c 2 -a 2 by /3 2 and y 2 , so that are the equations of the focal ellipse of the confocal system, we should have to put, with Jacobi s notation, = 7 2 cs 2 (u,,c), tf + \= , where and now u, v, w will be Lame"s parameters, as given in Max well s Electricity and Magnetism, I., chap. X. By solution of the three equations of the confocal quadrics, _ c 2 ct 2 . c 2 6 2 and thus x, y, z can be expressed as functions of u, v, w. Employing the function s a of 203, ms 22 , y 2 = OF THE ELLIPTIC FUNCTIONS. 273 When 6 2 = c 2 , the ellipsoids are oblate spheroids, and the hyperboloids of two sheets degenerate into planes through Ox ; and now the orthogonal system is given by 1 V 2 (i } cot% cec 2 u / w ^ii ^ y 2 z 2 y n - \ u -) ...(ui.) intersecting in the point x = y cot u tanh v, y = y cec u sech v cos w, z = y cec u sech v sin w. When b 2 = a 2 , the ellipsoids are prolate spheroids, and the hyperboloids of one sheet are planes through Oz\ now the orthogonal system is given by r~ + cechhi coth% + sechhv tanh% ~ ^ intersecting in the point x = y cech u sin v sech w, y = y cech u cos v sech w, z = y coth u tanh m The degenerate case of confocal paraboloids, where the centre is at an infinite distance, may be written 2/ 2 2 y 2 (viii.) intersecting in the point x = a(cosh u + cos v cosh w\ y = 4ta cosh J^ cos ^v sinh Jw, = 4a sinh ^u sin ^v cosh ^w. (Proc. Lond. Math. Society, XIX.) Q.E.F. 274 THE DOUBLE PERIODICITY 255. We may take u, v, w as Lame s thermometric para meters, and now Laplace s equation becomes (Maxwell, Elec tricity, I., chap. X.) Thus <j) = + 2Evw + 2 Fwu + 2 Guv + Huvw is a particular solution of this equation; for instance, the electric potential between two confocal ellipsoids, defined by U-L and it 2 , maintained at potentials U-^ and U% is given by When the solution <^> is equal to UVW, the product of three functions, U a function of u only, F of v, and W of w only, then Laplace s equation becomes ft = so that we may put three equations of Lame s form ( 204), when g = 256. The complete solution of Lame s equation was first obtained by Hermite, in the form Denoting by Fthe product U^U 2 of U^ and U 2 , or F(u) and F( u), two particular solutions of the general linear differential equation of the second order, in its canonical form where 1 is some function of u, and denoting differentiation with respect to u by accents, then or F / -2/F=2C7 1 / C7 2 ; and F //x - 2/F - 2/ r F- 2 [7/ CT 2 + 2 Z7/ f// or F // -4/F / -2/ / F=0, the general solution of which linear differential equation is OF THE ELLIPTIC FUNCTIONS. 275 A first integral of this differential equation is 2 YY" - F 2 - 4/F 2 + C 1 = 0, where C is a constant, given by U& -UJU^C, the integral of ^ J7 2 " - Uf U. 2 = 0. In Lame s differential equation and now, changing to x = pu as independent variable, and this equation for Fhas, as a particular solution, a rational integral function of x or jm, of the -nth order, which we may write F= and ^ Now, by logarithmic differentiation, ?7 2 ^ F Brioschi shows (Comptes Rendus, XCIL) that, when resolved into partial fractions, we may put pa) provided that and Then and, integrating, Fu, or [T, = n e X p( - fa) = II0(, a) ; while Z7 2 or J^ u) is obtained by changing the sign of or a, 276 DOUBLE PERIODICITY OF THE ELLIPTIC FUNCTIONS. 257. Hermite shows (Comptes Rendus, 1877) that the func tion F(u) may be otherwise expressed by / /I \ n-I *()=( and <f>u, called the simple element, is of the form e Xu cj)(u, w), </>(u, co) being a solution for n = 1 and h = poo ( 204). To obtain the coefficients A v A 2 , ... in F(u), we suppose <u or e Xu (p(u, (o), JF u, ^?u expanded in the neighbourhood of u = Q ( 195), in the form (Halphen, F. E. /., chap. VII.) Substituting in Lame s differential equation F"u= {n(n + I)pu + h}Fu, we obtain, by equating coefficients, _---- - 10 On comparing the two forms of the solution Fu, we find that w = 2a, and X = fo> 2fa. Thus, for instance, when TI = 2, we find, as in 209, d j( _ f(( _ When 7i = 3, w h e re ctj + a 2 + a 3 = a>, r l liis fails when g 2 = 0, and a 1 = v, a 2 = wf, a 3 = aj 2 ^ ; but now (229) J^ = i(^ CHAPTER IX. THE RESOLUTION OF THE ELLIPTIC FUNCTIONS INTO FACTORS AND SERIES. 258. The well-known expressions for the circular and hyper bolic functions in the form of finite and infinite products (Chrystal, Algebra, II., p. 322; Hobson, Trigonometry, chap. XVII.) have their analogues for the Elliptic Functions, as laid down by Abel in Crelle, 2 and 3. Granting the possibility of the resolution into linear factors, the individual factors are readily inferred from a consideration of the zeroes and infinities of the function. Denote 2mK+2nK i by ft, where m and n denote any integers, positive or negative, denote also Q + K or (2m + I)K+ ZnK i by Q v tt + K+K i or (2m + l)K+(2n+l)K i by Q 2 , and Q + K i or 2771^+ (2n + l)K i by Q 8 . Then considering the function sn u, the zeroes are given by u = fi, and the infinities by u = Q 3 ( 239) ; and thus we infer that, if sn 11 can be resolved into a convergent product of an infinite number of linear factors, the form is m = oo n = oo / n, \ u IT IT (l-j) " =-""=-< X (1) the accents in the numerator denoting that the simultaneous zero values of m and n are excluded. 277 278 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS Similarly, cnu = BTLn(l-) /D, .............. . ...... (2) ...................... (3) the zeroes of cnu being given by u = l v and the zeroes of dnu by u = } 2 , while the infinities are given as before by u = Q 3 ; D denoting the denominator in (1). 259. But now, in demonstrating the analytical equivalence of the expressions on the two sides of equations (1), (2), (3), it will fix the ideas if we employ a physical interpretation, such as that given in 247. It was shown there that the real and imaginary part (norm and amplitude) of log sn w, where w = u + vi, will represent in the rectangle OABG the potential and current function of the flow of electricity (or of liquid, following the laws of electrical flow) from a positive electrode at to a negative electrode at C, JTT amperes being the strength of the current ; but here we take OA=K, OG=K f ; and u, v are the coordinates of any point in the rectangle. The infinite series of electrodes, which are the optical images by reflexion of these two electrodes at and (7, will form a system on an infinite conducting plane, such that, if the strength of the current at each electrode is 2?r amperes, the resultant effect in the rectangle OABG will be the same as before. (Jochmann, Zeitschrift fur Mathematik, 18G5; 0. J. Lodge, Phil. Mag. 1876 ; Q. J. M. y XVII.) Starting with a single electrode at 0, of current 2?r amperes, the potential and current function at any point whose vector is w or u + vi are the norm and amplitude of logw; and logiv may be called the vector function of the electrode at 0. For an electrode at a point whose vector is c a-\-bi, the vector function 8&z=X+yi is log(0 c), which may be written disregarding the complex constant log(-c). INTO FACTORS AND SERIES. 279 The vector of any optical image of in the sides of the rectangle OABC being given by Q, the vector potential of the corresponding electrode is log(l w/Q); and the vector function of the system of images of the positive electrode at will be Similarly the vector function of the system of images of the negative electrode at C will be But these functions, considered separately, represent a physical impossibility, and are analytically meaningless ; their difference, however, will represent the vector function of the whole system of posi tive and negative electrodes ; and since this function satisfies the requisite conditions inside the rectangle OABC as the function logsnw, we are led to infer equation (1), with suitable restrictions explained hereafter. For log en w, the positive electrode is placed at A, the negative electrode being still at (7; the vectors of the positive electrode images are given by Q : ; and now equation (2) is inferred ; while for log dn w, the positive electrode is placed at B, and the vectors of its images are given by Q 2 , the negative electrode being at (7; and we infer equation (3). When in the rectangle OABC we have OA = a, OC=b, we take K lK^bja, and write K(x/a) + K i(y/b) for u + vi* x, y now denoting the coordinates of a point. 260. We now proceed to express these doubly infinite pro ducts of factors, corresponding to the different integral values of m and n, by means of singly infinite factors for different values of n ; that is, we combine all the factors for one value of n and the infinite series of values of m into a single ex pression; and here we employ the formulas for the trigono metrical functions expressed as infinite products. Interpreted physically, we determine the vector function of an infinite series of electrodes, equispaced on a straight line parallel to OA. 280 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS Denoting the vectors of such a series of positive electrodes by 2ma+nbi, the vector function is m= / _ _ log n (z-2ma-nbi), or iog(-n6i)IT(l- and provided that (z nbi)/2ma is ultimately zero when m is infinite, or that z/ma and n/m tend to the limit zero, we can write this vector function (Cayley, Elliptic Functions, p. 300) log sin \-w(z nbi)/a, ....................... (4) Resolved into its norm and amplitude, this vector function is i log J[cosh{7r(2/ nb)/a} cos TTX/O] + itan~ 1 [tanh{j7r(2/ ti&)/a}cot(j7ra;/a)]. ...(5) The amplitude or current function is therefore constant when a3=(2m-hl)a; and there is no How across these lines, provided however, as is physically evident, we do not recede to such a large distance from the origin that we are not justified in taking It z/2ma as zero. 261. We suppose that Oy passes through the centre of this infinite series of electrodes, or that m reaches to equal infinite positive and negative values; but now, at a very large dis tance from 0, the electrodes on one side of a line, given by o; = (2m-|-l)a, where m is a large number, will preponderate over the electrodes on the other side, and the resultant effect will be a uniform normal flow a across this line, to counteract which a term of the form az or loge~ az must be added to the vector function. The analytical equivalent of this physical effect is illustrated by the theorem proved in Hobson s Trigonometry, p. 328, that, when the integers p and q are made infinite in any given ratio, then 00, the limit of the product Ji) ... (1+ w V 2aA a/ \ a/\ 2a/ \ qa The infinite product TL(l+c n x) is convergent for all finite values of x, if the series 2c n is convergent ; as is evident on expanding the logarithm of the product. INTO FACTORS AND SERIES. 281 But Weierstrass shows (Berlin Sitz., 1876) that the divergent product can be made convergent if the exponential factor e z l ma is attached to the linear factor Iz/ma; or, interpreted electri cally, if to the motion due to the electrode at ma, whose vector function is log(l z/ma), we add a uniform streaming motion parallel to the vector ma, given by log e z l ma or z/ma. Now, denoting the harmonic series since the limit of s p log_p or s q logq is Eulers constant. 262. In a similar manner it is inferred that the vector function of an infinite series of positive electrodes, whose vectors are (2m + l)a-f-7i6i, m reaching to equal positive and negative infinite values, is log cos %Tr(z-ribi)/a = JlogJ[cosh{7r(y-w&)/a} + cos(7r#/a)] + i tan- 1 [tanh{ %Tr(y-nb)/a}t&n(^7rxla)], (7) having lines of equal amplitude given by x = 2ma. Therefore the vector function of a pair of lines of electrodes, whose vectors are 2manbi, is log sin{ \TT(Z nbi)/a }sin{ \ir(z -f nbi)/a} = log J{cosh(7i7r&/a) cos(7r^/a)} ; or, corrected by the addition of a constant, which makes the function vanish when z = Q, the vector function is , cosh(ti7r&/a) cos(7T0/a) , 1 2o w where q = e- 7rb/a . For a pair of lines of electrodes whose vectors are (2m + l)a?i6i, the vector function is which may be replaced by , cosh (ri7r6/a) + cos (irz/a) , 1 + 2q n cos(7rz/a) + (? 2n /0 ^ cosh(7i 7 r6/a)-|-l W2 For the line of electrodes along OA, whose vectors are 2ma or (2m + l)a, the vector function will be log sin(j7r/a) or log cos(|7T0/a) ................ (10) 282 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 263. Under Cayley s restrictions, that m reaches to equal positive and negative infinite values, and n also ; but that the infinite values of n are infinitely small compared with the infinite values of m (equivalent to taking the infinite array of the images of the electrodes as contained in an infinite rect angle, of which the length in the direction OA is infinitely greater than the breadth in the direction OB), we can now replace the doubly infinite products in (1), (2), (3) by singly infinite products, in the form , sn u = A sin(j7ru/A) II - r- v +D, (11) 71-1 I 1 "*? ) en u = B cos( JirM/JSr) II __L S -_ 4-D, (12) dnu= Oil where D= n By putting u = 0, the values of A, B, G are seen to be , 1, 1 ; while g = exp( - ^K \K}. The common denominator D of the three elliptic functions, which represents physically a function whose logarithm is the vector function of the negative electrodes at points whose vectors are of the form Q 3 , is the equivalent of Jacobi s Theta Function of 187; and we write 1 -00 nl 4 Li + The numerator of sn u will now be the equivalent of the Eta Function, defined in 192 ; and thus Hu = K sn u Qu . ...(16) The numerator of en u is represented by the Eta Function of u + K, and the numerator of dn u by the Theta Function of and the factors are so chosen that INTO FACTORS AND SERIES. 283 Equation (6) of 188 may now be written while, by means of (7), 137, H(+t;)H(u-t;)8*0=Hue 8 -eVBP (19) 264. It is convenient to replace \iru\K by a single letter x ; and we shall now find that the constant factors are so adjusted as to give the expansions in a Fourier series in the form o 9 a i /o/"^\ cin R/y {91\ sinox t^i; It is easily shown algebraically that 71=00 n=l by changing z into <fz and multiplying by qz, when the pro duct on the left hand side merely changes sign ; whence equa tion (20) is inferred from (15) by putting z = e 2xim } and equation (21) is obtained from (20)* by writing qz for z, and multi plying by q^z*. Written in the exponential form, >* (22) or with g = e~ a , a = 7rK IK, arid b = xi, n -^ b - ...... (24) and e(u+2K)= Ou, H(u + 2A r )=-Hu, .................... (25) Changing u into u + K i, or a; into x + ^i\ogq, we find iq-le-^eu, .............. (26) agreeing in giving K snusn(u + K i) = l, .................... .(27) and leading by differentiation to the formula Z(u + K i) = Zu + (cnuduu/suu)-(i7rilK), ......... (28) which, with ( 176), Z(u+K) = Zu-(K 2 snucuu/dnu), ..................... (29) leads to nu/cnu)-(%Tri/K) .......... (30) 284 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 265. Jacob! writes (Werke, I., p. 499) x for ^iru/K, and Ox for 9tt, OjX for Hit, ^ for H.(u + K), and # 3 ce for Q(u+K) , and now ......... (31) m ox ......... (32) m*if*-1W*~V* = 2q$cos x + 2^ cos %x -f 2^cos ox + ......... (33) = 2^V na:i ......... (34) or, with q = e~ a , b = xi, Ox = Ei 2n exp( - O^x = 2 exp ( (35) Conversely, starting with these functions as defined by these exponential series, it is possible to rewrite the whole theory of Elliptic Functions ab initio in the reverse order, and to deduce all the preceding results. (Jacobi, Werke, I., p. 499 ; Clifford, Math. Papers, p. 443.) For instance, we find that 6(x + JTT) - flgOJ, 0(x + \i log q)=- iq^e^x, O^x + JTT) - 2 x, O^x + \i log q) = - iq-$e xi 6x, 2 (x + ITT) = - 0^, 6 z (x + Ji log ? ) = g-V flgB, 3 (x + JTT) = 0c, 3 (o; + Ji log q) = q-iePOjc ...... (36) The quotient of two functions is thus a doubly periodic function, of reaZ period 2?r or TT, and imaginary period ilogq. The form of the and function series shows that they satisfy partial differential equations of the form _ dx z d log q and the functions are therefore suitable for the solution of problems in the Conduction of Heat. Thus, if 6(x cos a + y sin a, q) represents at any instant, = 0, the temperature at the point (x, y) of an infinite plane, of INTO FACTORS AND SERIES. 285 which y denotes the ihei^mometric conductivity, then at any subsequent time t, the temperature will be given by 0(o; cos a + # sin a, qe -W) .................... (38) 266. Similar considerations to those of 258 enable us to resolve other expressions into factors ; for instance, ., . , dnu-f/ccnu , or its reciprocal so that K K dn u KCIIU K /dn u K en u Vdnu K dn u -f- K _ A/ , K dn u + K en u \ dn u + KCHU Now dc u, or sii(Ku) = l//c, when or cos \TTUJ K= cosh(2r& - 1) while dc u = I/K, when cos %TTU/K= cosh(2w V)^ and therefore we may put dn u K en u _ _.cosh(2?i V)\TrK ]R cos \irU\K , 11 n - -" - 1J where the letter C is used to denote some constant factor. Now, writing x for ^TrujK, and supposing x and it real, log(l - 2c cos x + c 2 ) = log(l - ce xi ) + log(l - ce-**) = 2(c cos x + Jc 2 cos 2a; + Jc 3 cos 3x +...), log(l + 2c cos cc + c 2 ) = 2(c cos x Jc 2 cos 2ic + Jc 3 cos 3a: ...), log = - -5 = 4(c cos a; + ic 3 cos 3x + |c 5 cos bx +...). 1 + LC cos x -f- c - Therefore, expanding the logarithm of (39), log- l ^- = logC- = \ogC- ,l-q~ 3 1-53- 5 !_ 5 5 yv AV 1 cos(2??i l)^7ru!K = \ogC 22 r- -. , ^ , and, differentiating, snu=S , ,..(41) JL sinh2m the expression of sn u in a Fourier Series. 286 TH E RESOLUTION OF THE ELLIPTIC FUNCTIONS 267. By forming the similar factorial expressions for Ksnu+iduu and suu + icnu, and taking logarithms, we shall find log(/c sn u + i dn u) .^, 1 sin(2m = constant SiS 2m -1 eosh(2m , , , . N .^1 sin rri log( sn u+^ en u)= constant ^E r t, , r ~ ...... (43) m cosh m-TrK IK and, differentiating, ?r^ cos(2m TT and therefore, integrating, We have now found that, in 78, 1 n cosh 268. From 263, we find, in a similar manner, that 7T 2 Now, referring back to 78, we can put ^ 7T 1 7T 2( TTX _, sn - am u = -x-^4- 2 r -- WTTT ................ (46) m cosh m-jrK IK v l c = constant 2, -- ^TT -- 7^771^; .................. (47) m Binh(mxJi 7 JT) and, differentiating, 7r s ^iKm7rV^_ -^ J " (49) K Kl-q^ Putting u = in (49) or (50) gives what is called "a q series," m toiuf* _K(K-E) //77-\ ^ i INTO FACTORS AND SERIES. 287 As an exercise, the student may form the similar factorial expressions for 1 cnu 1 snu 1 dnu duu cnu etc. sn u en u- K sn u yc sn u and their reciprocals 1-fcnu 1 + snu 1 + dnu dnu + snu , sn u CD u K sn u jc sn u and thence determine, by logarithmic differentiation, the Fourier Series for ns u, cs u, ds u, etc. (Glaisher, Q. J. M., XVII.). The applications of these expansions will be found in papers in the Q. J. M., XVIIL, XIX., XX. 269. As an application of these q series, consider the problem of the electrification of two insulated spheres, in presence of each other, of radii a and b, and at a distance c from centre to centre, when maintained at potentials V a and V b , with charges of E a and E b (Maxwell, Electricity and Magnetism, I, chap. XL). Then E a = q aa V a + q ab V b , E b = q ab V a + q bb V b , (52) where q^, q bb are called the coefficients of capacity, and q^ the coefficient of induction. We take u and v as coordinates, given by the dipolar system x+yi = kt&u^(u+vi), (53) so that u = constant represents a circle through the poles (0, k), and v = constant represents an orthogonal circle, with the poles as limiting points. Now, if we revolve this system about the axis Oy, which may be supposed vertical, the two spheres, if outside each other, may be supposed defined by v = a and v= /3, so that a = fc cosech a, b = k cosech /3, c = /j(cotha + coth/3) ; and putting a + /3 = aJ, Maxwell shows, by Sir W. Thomson s method of successive images, that q aa = kZ cosech (n?3 /3), q a b = &2 cosech nft, q bb = k2 cosech(?iT a), (54) the summations extending for all positive integral values of n from 1 to oc . Here q ab is called Lambert s Series ; it is considered in the Fundamenta Nova, 66. 288 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS Again, with a ft = x, = k2 cosech and by the preceding formulas it can be shown that Trr/ ................ (55) When the two spheres are equal, x = Q, and = = k% cosech Zn - When /3 = 0, the sphere ft becomes a plane; and now q aa = q ab = /c2 cosech Tia = a sinh aS cosech net ; which shows that the capacity of a sphere of radius a is raised from a to ct sinh aS cosech Tia by the presence of an uninsulated plane at a distance a cosh a from its centre. Similar functions occur in the determination of the motion of two cylinders or spheres, defined by v = a and ft, when the interspace is filled with homogeneous frictionless liquid. (W. M. Hicks, Phil Trans., 1880 ; Q. J. M. t XVII, XVIII. ; Basset, Hydrodynamics, I, Chaps. X., XL ; C. Neumann, Hydrodynamische Untersuchungen. ) 270. To illustrate geometrically the singly infinite product forms in 263 of the elliptic functions, consider the analogous problems of electrodes at the corners of curvilinear rectangular plates, bounded by arcs of concentric circles and their radii. The vectors from the centre as origin of a series of p electrodes, equally spaced round a circle of radius a, will be aexp 2r?ri/p, where r = l, 2, 3, ..., p\ and with polar coordinates r, 0, the vector of the point will be T exp i6 ; so that for the p electrodes, each conducting a current of 2?r amperes, the vector function is loglfjr exp(i#)-a exp(2r7ri/p)} =log(r^e ? ^-a^), ..... (56) by De Moivre s Theorem (Hobson, Trigonometry, Chap. XIII.). Interpreted geometrically, the norm is the logarithm of the product of the distances of an} 7 point P from the electrodes, while the amplitude is the sum of the angles the lines joining the electrodes to P make with the vector $ 0. INTO FACTORS AND SERIES. 289 We thus prove incidentally one of Cotes s theorems, namely. that the square of the product of these distances is ( r p e ipe - a P)(rPe ~ W -aP) = r 2 ? - 2a*r*cos p6 + a 2 ^, ... (57) and, in addition, the theorem that the sum of the angles the vectors from the electrodes to P make with the vector $ = is r^sin pQ tan 1 ; ..................... (08) a p and when the sum of these angles is constant, the locus of P is an oblique trajectory of the curves rPcospO or r^sin pO = constant. With a single negative electrode at the centre, of current nw amperes, half the total current from the n electrodes on the circle will flow to 0, the other half flowing off to infinit} . Now the vector potential is, on writing e p for r/a, j. r n sin n i tan 1 - r n cosnO a n We can isolate a sector, bounded by = 0, 9 = ir/n t and / = a; and the preceding expression will represent the vector function of the electrical flow of JTT amperes, with electrodes at the end of the vectors r = a, and at r = 0. The amplitude of this expression will also represent the temperature in this sector, if the radius 6 = is maintained at temperature 0, while the radius (9 = 7r/?i and the arc r = a are maintained at temperature |TT. 271. Now suppose that on the same circle r = a, an equal number p of negative electrodes are placed, equally spaced be tween the positive electrodes ; the vectors of these electrodes being a exp(2?^ l^i/p, the vector function is or, if moved out radially on to a circle of radius b, -log^-PeW + bP) ......................... (60, The vector function of p equal electrodes at a exp ZTTTI/}), and of p equal negative electrodes at a exp(2r l)-7ri/p will therefore be log^e?* 6 aP)/(rPe ip6 + a*) ; which, when resolved into its norm and amplitude, is --H tan~ : - 7-* -t-za r* cosptJ-|-a** J.E.F. T 290 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS , . cospO , . , , sin pO = tanh 1 r^- 4^ tan" 1 . .......... (61) with p = log(r/a) ; this function will represent the state of electrical motion in a wedge bounded by $ = and Q = Trjp. 272. The substitution in the preceding expressions in 24-7 of the conjugate functions pO and log(r/a)^ or pp for u and v, leads to the solution of corresponding problems for curvilinear rectangles bounded by arcs of concentric circles and their radii ; and now q=(b/a) p , where a and b are the radii of the curved sides, while tr/p is the angle between the straight radial sides ; so that in the rectangle OABC, OA = a7r/p, BC=b-7r/p, OC=AB = a-b. The vectors of the imaes of an electrode at are now where n denotes any integer, positive or negative, and r = l, 2, 3, ...,n. For electrodes at A, B, C, the vectors of the images are For a given value of n, the vector potential of the electrodes, whose vectors on a circle of radius aq nfp are aq^Pexp^riTr/p or aq nlp exp(2r ty-jri/p will be \ogU(r^ d -aPq n ) or logU(r p e ipd + a p q n ) .......... (62) Now, suppose a positive electrode is placed at and a negative electrode at C, with the corresponding system of images ; the vector function is low "i on introducing a negative electrode, of current TT amperes, at the origin ; and, writing irW/K for p6 + ilog(a/r)P, this becomes INTO FACTORS AND SERIES. 291 equivalent, as in 263, on omitting constant terms, to log sn iv. A similar procedure with electrodes at A, C, and B, C, will lead to the singly infinite factorial expressions for en it, and dnu. Projecting these equipotential and stream lines stereographi- cally on a sphere which touches the plane, we shall obtain the corresponding solutions for the flow of electricity on the surface of the sphere. (Robertson Smith, Proc. R. S. of Edinburgh, vol. VII. ; M. J. M. Hill and A. J. C. Allen, Q. J. M. y XVI, XVII.) 273. When these electrodes are replaced by straight parallel vortices, perpendicular to the plane, which is taken as hori zontal, the potential and stream functions are interchanged. Suppose a vortex is placed at a point P in the rectangle OABC ; to introduce the restriction that there is no flow across the sides of the rectangle, we must suppose the motion due to vortices which are the optical reflexions of the point P in the sides of the rectangle ; the sign of the vortex being positive or negative according as the corresponding image has been formed by an even or odd number of reflexions. The vectors of the positive images will therefore be and of the negative images 2ma + 2nbi z ; wh ere z = x -+- y i- , .- = x yi. The resultant current and velocity function at =+>;* will therefore be the norm and amplitude of _ ( 2ma + 2nbi + -z )(2ma + 2nbi + {;+z )" At the point P, this vector function, due to all the other images, is therefore ( 2 ma -f 2nbi + z z and writing r,- = -> an d 2l^ K a a b this may, according to 263, be replaced by 292 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS The stream function at P is therefore, disregarding constants, 9% H 2 w- H% 2 w , v = | log(ns 2 K, ns 2 m) = J Iog{ns 2 (u, /c) + ns 2 0, * )-!} : ...(66) so that the curve described by the vortex is given by ns 2 (2 Kx/a, K ) + u/(2K y/b, K) = constant, ......... (67) and all the other image vortices keep up a symmetrical dance, by describing similar curves. 274. The vortex is stationary when at the centre of the rectangle; and now, changing to the centre as origin, the vectors of the images are ma+nbi, where m + 7i is even for the positive, and odd for the negative images; so that the vector function of the motion is given by , n ,, cnw //% . v = log =Jlog T . ................. (68) en kw 1 + cnit; Expressed as norm and amplitude, as in 247, this function _ i I 1 en w 1 en w l 1 en w 1 -f en w 4 cn^// ,, cuvi cuu ,, snu dn vi = ^ log ---- r- - + 1 log - & . , . . en vi + en u sn u dn vi + dn u sn w , , en u , , sn u dn t i tanh ~ J . tanh ~ . - T - . en w dn u sn w . 4 .snttdn t //<r ., en v) + ^ tan -1 , ........... (09) dn u sn ?? with u ZKxja, v = 2K y/b; the modulus of the elliptic func tions of v being /. The equation of a stream line of liquid is therefore given by en u en v = constant, or cn(2Kx/a, K )cu(2K y/b, K) = constant ............. (70) Close up to a vortex the velocity according to these ex pressions would become infinitely great, which is physically impossible; but a solid core may be substituted for this central portion, and the shape of this core has been investigated by J. H. Michell, Phil. Trans., 1890. INTO FACTORS AND SERIES. 293 275. When a point is placed inside an equilateral triangle, the Kaleidoscopic series of positive images is given by the vectors z, wz, w 2 z, where z = x + yi, and w is an imaginary cube root of unity ; the negative images being given by z, coz , o)V, where z = x + yi ; the origin being at a corner of the triangle, and the axis of x perpendicular to the opposite side (Fig. 27, i.). (i.) Fig. 27. (ii.) In addition, similar groups of six images must be added, ranged round the centre of hexagons forming a tesselated pave ment, the vectors of the centres of the hexagons being 2mh + 2nhij3 and (2m + l)/i + (2?i + 1)^^/3, where h denotes the altitude of the equilateral triangle. In the corresponding doubly inh nite products, the elliptic func tions will have K jK= l J% i so that ( 47), /c = sin 15, 2or = J. Then, in Weierstrass s notation, the vector potential at for a single source or electrode inside the triangle will, neglect ing constant terms and factors, be expressed by ( 278) lo g o- (-s V (f-fttf )(r (i-orz) (71) while for a vortex or electrified wire, the vector potential is The nature of the resolution of these functions into their norm and amplitude is illustrated in 227 to 231. (O. J. Lodge, Phil. Mug., 1876; 0. Zimmermann, Das logar- ithmische Potential einer gleichseitig dreieckigen Platte, Diss. Jena, 1880 ; A. E. H. Love, Vortex Motion in Certain Triangles, Am. J. M., XI.) 294 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS So also for a rectangular boundary OACB, if we write a for g-x + (r]-y)i, or -2, ft for g+ x + (i-y)i, or +/, y for +a + (>? + 2/)i or f+z, 5 for ^-a + fo + Z/)*, or faf; 3;, -2 , 0, 5? being the vectors of the point P and its images by reflexion in the coordinate axes Ox, Oy, taken in order in the four quadrants ; then the vectors of all the other images by reflexion in the sides of the rectangle OABC being ranged in a similar manner round points whose vectors are 2ma-f 2?i6i, it follows from what has gone before that we may express the vector function at f of all their images, taken as positive, by log era v/3 o-y rS, ........................ (73) with w l = a, a) 3 = bi ; disregarding constant factors, and exponential factors of the form exp(Au + Bu 2 ). But when we represent the vector potential of a vortex or electrified wire at P, the vector potential becomes 276. As another illustration of the connexion of a regular Kaleidoscopic figure with Elliptic Functions, consider the solu tion of the reciprocant 8 = 0, .................. (75) dii d 2 ii 7 d s y d*y where ^ 8B 7 i a = j v ^="T^> c= i dx dor dor dx* (Sylvester, Lectures on the Theory of Reciprocants, VI., 1888.) Mr. J. Hammond has shown (Nature, Jan. 7, 1886, p. 231 ; PTGC. L. M. S., XVII., p. 128) that the integral of this equa tion (75) may be written (l + ti)dt ( n > By turning the axes through an angle ^tan^A/ic), we can make X vanish ; and now, replacing JAC by unity, ^-^5 0,4),. ..(78) and K* + y*)K-2^)= 1 ......................... ( 79 > INTO FACTORS AND SERIES. 295 Since (196) $>(joz = copz, $>a) 2 z = a) z pz, where eo is an imaginary cube root of unity, therefore pu(x+yi)&u?(x-yi) = i, .................... (80) which shows that the curve is unchanged if turned through an angle of 60 about the origin (Fig. 27, ii.). Captain MacMahon has shown that the intrinsic equation of this curve may be written cos3^ = cln(s/c), with jc=JV 2 ................ ( 81 ) The student may also show that the equation of the curve may be written in one of the forms K 2 sn 2 (#, K) = ic /2 sn 2 (y, AC ), K)dn(y, K ) = K, .............................. (82) with /c = sinl5, /c = sin75. As a similar exercise, the student may solve the rcciprocant fc-56 = .............................. (83) in the form $>x $>y = -1, ............................ (84) and determine its intrinsic equation, drawing the correspond ing curves (Proc. London Math. Soc., XVII., p. 360). 277. When we expand, in ascending powers of u, the logarithm of a doubly infinite product, such as that in the numerator of sn u in equation (1), 258, we find Now, when the origin is taken at the centre of all the points whose vectors are Q, the coefficients of u, u 3 , u 5 , ... vanish ; but the value of the series is still indeterminate, until the infinite curve containing all these points has been defined. For if P denotes this infinite product, and P its value when the boundary has changed into a similar curve, then where the summation now extends over the region lying be tween the two boundaries; and now the limit of SQ~ 2 is a definite number, A suppose, while the limit of 2Q~ 4 . ... is zero. Therefore logP / -logP=.Wu 2 , or P = Pe* Au *...-, ......... (86) so that the value of the infinite product depends on the shape of the infinite boundary (Clifford, Math. Papers, p. 463). 296 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS But, as in $ 261, Weierstrass removes this ambiguity \yy attaching to each linear factor of the product, such as ,. i f (u 1 an exponential factor exp( c -f - \l2 z and, in the physical analogue, the corresponding electrode at 12, whose vector function is log(l u/Q), must have associated with it a uniform flow in the direction of the vector Q, repre sented by u/Q ; and a streaming motion iri rectangular hyper bolas, whose asymptotes are parallel and perpendicular to the vector Q, represented by |(u/Q) 2 . Now in the expansion of the logarithm of the doubly infinite product P, when these exponential factors are introduced, logP = logu-XZQ- 4 -iu 6 Sft- 6 -..., ........... (87) an absolutely convergent series ; that is, a series the value of which is independent of the order of the terms. 278. Making a new start ab initio with the sigma func tion ( 195), as defined now by the equation n/ TT/ It > - /TT\ n l-exp+5 , ......... (U) where Q = 2mto + 2iico , and ta /wi is a real positive quantity, so that co, a) correspond to w> v co 3 or co 2 , a) 2 according as A is posi tive or negative, then a-u is the analogue of Jacobi s Eta Func tion ; i n fact, a-u = Ce Au R^(e l - e s )u = Ce^O^TrU/u), ....... (88) ( 263), where C, A are certain constants ; also log a-u is the same as logP in equation (87). Now denoting, as in 195, d log o-u , - , cZ 2 lo(r aril du ~ by ^ and - or (V) by differentiation of ( U) and (58) ; so that, on reference to 195, we may put # 2 = 602Q- 4 , # 3 = 14.0ZQ- 6 , ............... (W) also 2 = 2 4 .3.5 2 .72Q- 8 , = 2 4 . 3. 5 . 7. 11 2Q 10 , etc. INTO FACTORS AND SERIES. 297 Differentiating (60) again, p tt= -- r (,7Q? ......................... (Y) Then (o-u)/u, ufu, u 2 ^u, u 3 ^ u, u*p"u, ..., are unaffected by the considerations of homogeneity of *196; as for instance in the expansions in equations (21) and (22) on p. 249. A change in (X) and (Y) of u into u + 2pw + 2qw, where p and q are integers, merely leads to a rearrangement of terms ; so that, as in 250, p( U + Ipw + 2^0) ) = pit. Also, since in Q = 2m&&gt;-f2?7a> , the arrangements (771, -?i) and ( ?7i, ri) exist in pairs, therefore and p /2 u = 4 . pw, p w . pu, = lp*u-g 2 pu-g 3 , ................................. (AA) as originally defined otherwise in 50. A change of u into u + 2w in (V) shows that, by a rearrange ment of terms, ftw + 2ft>) = 6 + 217, ..................... (89) where q is a certain constant, determined by putting w= w, so that / = & ............................ (90) Similarly ^ + 2o/) = fu + 2i/, ..................... (91) where ? $ ; .......................... (92) and, generally, f(u + 22^ + 2go) / ) = t + 2p, ; +2^ / ............ (BB) Integrating (^9) and (90), <r(u + 2w) = Ce*i<ru,, <r(w + 2o) ) = WVw ; where (7 and (7 are determined by putting u = oo and a/ ; so that (7(u + 2o))= _ e *+)(rM, o-( + iV)= -e*^ M + w Vit, (93) and therefore ,, ..................... (94) , ..................... (95) and, generally, (f(u + 2^0) + ^gco ) = - ( - 1 )(P+lX?+l) e C2,^+2 9 7 7 Xt 4 +; )w + 9 a; ) (rM/) . . . (CO) obtained also by integration of (BB). 298 TH E RESOLUTION OF THE ELLIPTIC FUNCTIONS The doubly infinite products in (U) may be converted into singly infinite products ; and now where q = e l", and trie* = ix 2 - 7r 2 S , j = J-TT* - 7r 2 2 cosech 2 (W/a>;), ... .(97) etc. ; for the proof of these and other similar formulas merely stated here, the reader is referred to Schwarz and Halphen. Also, denoting Q + w, ft + + , Q + w by Q x , fi 2 , Q 3 , then the function cr a % of 202 may be otherwise defined ab initio by the relation ^ ^ = ^Iin(l-)exp(J + Ig), ........ (EE) which will be found to lead to the preceding results. d 2 Denoting j- 2 logout* by <p a u, we shall find that &y* = K + a>a)> = 1 2 > 3 .................. ( 98 > (A. R. Forsyth, Q. J. M., XXII.) 279. Returning to the function G of equations (8) and (10), 215, and changing the sign of the us, we may also write it _ cr(v + Ui + U 2 + + UH)<T(V ~ UI)<T(V U 2 ) - . . <r(v u u ) (99) and since we may suppose the it s and v to be all increased by equal amounts, the condition (9) of 215 is no longer required. Now, since G vanishes when v = u r , where r=l, 2, 3, ..., /x; therefore the coefficients c , c v c 2 , ..., C M are determined by a series of equations of the form = c o + c i^^ + c 2F / u r+...+c^- 1 )it r ; ......... (100) and therefore the determinant (101) where M is a factor independent of v ; and now this theorem, as a corollary of Abel s theorem, shows that the determinant also vanishes when v= u-^ u . . . u^. INTO FACTORS AND SERIES. 290 The symmetry of the determinant shows that M must be a symmetric function of the us ; or writing u for v, and denot ing the determinant by <j>(u Q , u lt u. 2 , . . . , Up\ then cp is a symmetric function of the it s, such that . x _ A u . . . i - and it will be found (Schwarz, 14) that J. =(-1)^-1)1! 2! 3! .../il. Thus, for instance, with JUL = 2, 1, pw, p zi, =2 1, pv ^ By forming a similar function C" of the u"s, subject to the condition (6) of 215, we see that (7) is an elliptic function of v, which can be expressed by C/C , where C and C f are given by determinants, as above. Equation (CC) is also sufficient to prove that the function in (7) 215 is doubly periodic. As an application of the principles of this article and of 209, 215, 216, 257, the student may prove that Q of 215 is, writing a for u v b for u.- and u for v, given by the equations )o-(u + b)ar(a + b) <r(u + a -f- b)a-u a-a orb 1 , pa, 1, pa, p a 1, p6, p ft We thus verify the equations of 209, 257, du = $(u, a)<t(u- t b). When condition (6) of 215 is not satisfied, then (7) reappears qualified by an exponential factor of the form e pv when v is increased by 2>o) + 2go/; the function is then called by Hermite a doubly periodic function of the second kind ; the function (p(u, v) defined in 201 being the simplest instance of this kind of function. 300 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 280. Making the i/, s all equal, as in 218, and interchanging u and v, the function rfu+ftf is a doubly periodic function which can be expressed in the form of C ; but now the coefficients c must be determined by a series of equations of the form Expressed as a determinant we may now put pu-pv, Finally, making u = v, and dividing both sides by we find, in the limit, l=M * "" where Halphen denotes this function of u by ^(p+ Thus for instance, as in 200, with /* = 1, = " ~2 (Schwarz, 15); . t =-pu. Again, with /x = 2, By logarithmic differentiation, lo ^n^ = -- lo , = 7i 2 ^ - whence $nu can be expressed rationally in terms of When u = v, , . . . .(HH) , $> u, ____ INTO FACTORS AND SERIES. 3Q1 Also, when u = 0, = a M (-l>* + V ; ............................ (102) and therefore a^ = 0, when jj.v = Sp^ + 2<?o> 3 . 281. In the pseudo-elliptic integrals ( 218) yuv = (mod. w v o> 3 ); and now, knowing the number /z, the coefficients c , c v &,, ... in C or xu are readily calculated from a knowledge of the values of py, p v, p"v, ... ; in this way the results employed in 218, 219, 223, 225, 233 were inferred. Thus, for instance, in 219, we know that JUL = 3, JULV = 3a) x + o> 3 ; pv = J, p v = 3V 2 > P^ t= -6, p" v = 18i^/2, >"% = -252, ...; so that the ratios of c , c p c 2 , . . . can be calculated from the equations = c + ^4- 3i^/2c 2 6c 3 , 0- -6^ + 18^2^- 252c 3 . Taking an arbitrary value of c 3 , say f, we find, by solution, V=-9, c 1= -10, c 2 =-3V2; ^ = f c s (f p"u 31^2 p w, 10 j?u 9) = fc 3 {(2 p Now u so that, in the algebraical herpolhode referred to axes rotating with a certain angular velocity, we may put thus leading to the results of 219. As other numerical examples the student may investigate the results of 218, 223, 225, 233 ; also the example due to Abel (CEuvres, I, p. 142), where yu = 5, # 2 =12, # 3 = 19, and v = f w 2 or ift> 2 , when <@v = 2 or 1 ; we then find that the values of c , c v c 2 , c 3 , c 4 , c. are proportional to -288, -36, -48^3, 12, ij 3, 0; or -396, -252, -12i^/3, -24, ^3, 0. 302 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS Writing s for $u, then we may put X u = - 288 - 36#m - 48i^/3p ie, + 1 2%>"u + i J 3p"u = 36(2s 2 -s- 10) + 12i x /3(s-4) /v /(4.s 3 -12s- 10), M = - 396 - 252w- We thence infer that the corresponding pseudo-elliptic inte grals involve _ 1 ._ . 2(8-1) . tan -_ > ~ and now by differentiation we infer that 2s + 13 cs_ 2_ _ ,(8 -4)^/^-1^-1 9) !^" s __ " * ~ /" r> " x-*. ~t y-v \ "" /-A Ldll S 1 J s + 2 Thus, in the Weierstrassian notation, = - tan ~ V)W <0V with f/ 2 = 1 2, # 3 = 1 9, according as yv = 1 or - 2. These results may be employed in the construction of degenerate cases of the catenaries discussed in 80, 205, 206. Thus, for instance, the curve given by r ^o^( 2^/3u-oO~) = ^/^ is a plane catenary for a central attraction ri*wr per unit of length, in which ( 80) So also a tortuous catenary is given by the equations / / 5 cos(5$ + under an attraction nhvr to the axis Ox. INTO FACTORS AND SERIES. 303 282. Other pseudo-elliptic integrals are formed by the sum of two or more elliptic integrals of the third kind, when the sum of the parameters is of the form pw + qco, as in 226, for the expressions of and . We shall denote the integral of the third kind in the form (fa), 199, by $(u, v), as this we have found is the form of most frequent occurrence in the dynamical applications ; and now (fa) shows that $(u, a + b) ,11 <r(a u)<r(b +Mog-7 i \ 8 <r(a + u)o-(b + IQ pa p by reason of (y), 197, and (K), 200 When a + 6 = a> a , p ( a + 6) = 0, $(tt, a + 6) = ; and now , 6)= - By equation (N), 249, we may write i lo<y g(a + .)- g q = tanh , 1 /Am-e a . pa-q, . pq- p p(a u) e a \ \pa ea.pa ep.jpu , , p a pu e . . / /a pu = tanh- J , , or % tan- 1 f pa e a the latter form to be employed in dynamical problems, where p a is always imaginary ; thence the expressions given for f and in 226 can be inferred. As an application we can put a + b = a) l + (a s or co 3 in 209, and thence deduce a degenerate case of the Spherical Pendulum. EXAMPLES. 1. Prove the following q series : (i.) ... ... 90 (iv.) (1- (v.) ^/(KK)^. 2q^, g^ T V/cV 2 , J"sl/l72S</ 2 , or l/1728g, accord ing as A is positive or negative, when q and K or // is small. 304 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 2. With the notation of g 265, prove the theorem = 2<9 1 <6> 1 (6- - y - z)0 l (s -z- x^s -x-y}, where 2s = w+x + y + z. Deduce the formulas (i.) A /2 sn u sn v sn r sn s Ac 2 cn u en v cri ? en s + dn u dn v dn r dn s K Z = 0, provided u + v -\-r-\- 8 = 0. (ii.) K 2 sn -J(u -}- v + r + s)sn | ( & + v r s) X sn ^(u v + r )sn J(w v r+s) 3. Show that = 2(e 2 - e s ) (e s - ^(^ 4. Show that Weierstrass function a(u) satisfies the partial differential equations Show that the second of these equations is also satisfied by the function cra(u)/{ (e a - ep)(e a - e 7 ) }i ; and write down the differential equation satisfied by o- a u. 5. Prove that the projection of a geodesic on a quadric of revolution on a plane perpendicular to the axis is analytically similar to a herpolhode (Halphen, II., Chap. VI.). 6. Evaluate the surface of an ellipsoid. 7. Construct some degenerate cases of trajectories or caten aries on a sphere, or on a vertical paraboloid or cone, employing the numerical results of the pseudo elliptic integrals. CHAPTER X. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 283. By the Theory of Transformation is meant the ex pression, in terms of the elliptic functions of modulus K and argument u, of an elliptic function with respect to a new modulus X and of a proportional argument u/M; and then M is called the multiplier, and the relation connecting the moduli X and AC is called the modular equation. A particular case of Transformation has already been intro duced in Landen s Transformation ( 28, 67, 71, 123, 181, 182) in its application to Pendulum Motion, and to the Rectification of the Hyperbola. In accordance with the plan of this treatise, we begin with a physical application of the Theory of Transformation, before proceeding to the analytical treatment of the subject. Suppose then in 259 that an odd number, n, of such rectangles as OABC are placed in contact, side by side, so as to form a single rectangle OA n B n C, of length OA n = na, [and height 0(7=6 ; and now put OA /OC= a/5 = A/A , so that A /A = nK IK , ........................... (1) where K, K denote the quarter periods with respect to the modulus K ( 11), and A, A with respect to the modulus X. Let us begin by placing a positive electrode at 0, and an equal negative electrode at (7; then, inside the rectangle OB, the vector function will be log sn Az/a = log sn(Ax/a+A iy/b), with z = x + yi. G.E.F. 306 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. But, inside the rectangle OB n , the vector function of these electrodes and their images will be that due to positive elec trodes at 2sa and negative electrodes at 2sa+bi, where s assumes all integral values from to n 1; and the vector function of this system is ( 259, 275) s = n-l log II sn K(z-2sa)/na = log ILsu(Kx/na+K iy/b- 2sK/ri). s = Q The physical equivalence of these two forms of the vector function, as seen from two different points of view, shows that or sn(u/lf, X) = A li sn(u 2sK/n), (2) where u/M= Az/a, u = Kzjna ; so that M = K/nA = K /A ; (3) this is the formula for HIQ first real transformation of the sn function, of the nth order. Similar considerations will show that >, (4) (5) If, as in 263, we put q = exp( TrK jK), and r = exp( TrA /A) ; then r = q n , (6) and X is less than /c. It simplifies matters to place the rectangle OB in the middle of n such rectangles placed side by side, and now s ranges from ^(n 1) to J(?i+l); and combining equal posi tive and negative values of s, we find, according to (7) 137, =!(- 1) sn 2 ?y, sn 2 2s TT OiJL W/ O1JL o(jU H s= i JL rr where co = K/n ; oi> y=^ n i 1 rXv y (8) JjfJ. .L ^^ /C (JL w connecting y = sn(u/M,\) and x = su(u, /c), a = sn(2sK/n). 284. Next suppose that w equal rectangles, such as OABC, are piled on each other, so as to form a single rectangle OAB n C n , where OA =a, OC n = nb ; and now put THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 307 OA/OC = a/6 = A/A ; so that K IK=nA!/A ..................................... (9) The physical equivalence of a positive electrode at and an equal negative electrode at C, and of their images in the rect angle OABC, with the positive electrodes at 2sK iy/b and the negative electrodes at (2s+l)K iy/b in the rectangle OAB n C n and their images, shows in a similar manner that sn( Az/a, X) = A II sn(Kx/a + K iy/nb - ZsK ijn), where s may assume all integral values from to n l, but preferably, from J(??, 1) to i(?i + l); or sn(u/M, X) = A II sn(tt - ZsK i/n, K ), ........... (10) where u/M Az/a, u = Kzja ; so that M=K/A = K /nA -, ..................... (11) and now, with q = exp( TrK jK), r=exp( xA /A), we have r = q 1 / 1 , .............................. (12) and now X is greater than /c. Similar considerations show that, by placing positive and negative electrodes at A and C, or B and C, we shall obtain the formulas cu(u/M, \) = BTL cu(u - 2sK i/n) ; ............ (13) du(u/M, \) = CIL dn(u - ZsK i/n) ; ............ (14) these are the formulas for the second real transformation of the elliptic functions, of the Tith order. A similar physical interpretation of Transformation may be given in connexion with the curvilinear rectangles bounded by concentric circular arcs and their radii, as discussed in 270. 285. Besides the first and second real transformations in which q is changed into q n and q lin , now denoted by r^ and T O , there are in addition n 1 imaginary transformations, when n is a prime number, in which q is changed into w p q l/n , denoted by r p , where p = 1, 2, 3, ..., n 1, and o> is an imaginary Tith root of unity ; so that, corresponding to a given value of K, the modular equation of the ?ith order, if prime will be of the (?i + l)th degree in X, having the roots ^oo \) ^l ^2 ^n-l> of which two only, X and X , will be real ; \ x < K < X . 308 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. We need only consider the Transformations of prime order, as a Transformation of composite order, mn, can be made to depend on the transformations of the mth and nth order. The different transformations of the mrith order are formed by changing q into <? m/n ; so that the number of transformations for any number in general is the number of divisors of mn ; reducing to ?i + l, as before, for a prime number n. For a transformation of order ri 2 there is one real transforma tion for which q remains unaltered, and we thus obtain the formulas for Multiplication of the argument u by n. 286. After this physical introduction, we can proceed to the general algebraical theory of Transformation, as developed by Jacobi in his Fundamenta nova theories functionum ellipti- carum, 1829. The theory in its generality consists in the determination of y as a rational algebraical function of x, of the form y=Uir, ............................. (15) where U and V are rational interal functions of x, so as to satisfy a differeotial relation of the form Mdy dx where X= ax* + 4bx*+ 6cx 2 + dx + e, \ Y=Ay* + 4,By s + 6Cy 2 + 4<Dy + E,)" Making the substitution of (15), we find that we must have dx and the first condition requisite is that where T is a rational integral function of x, of the (2n 2)th degree ; and now, if we can make (20) where If is a constant multiplier, the Transformation is effected. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 309 But if U and V are both of the nth degree, or if one of the nth and the other of the (n l)th degree, so that either a n or b n (not both) is zero, this is necessarily the case ; for any square factor in (U, F) 4 will appear as a linear factor of dU n dV j V U =j j dx dx which is also of the (2n *2)th degree, and can therefore only differ from I 7 by a constant factor M. The Transformation is now said to he of the nth order. By taking X of the sixth, instead of the fourth degree, Mr. W. Burnside has derived hyperelliptic integrals (Proc. L. M. S., XXIII.) from the elliptic element dy/*JY, similar to the hyper- elliptic integrals of 159, 160, by means of substitutions of the second, third, and higher orders. Now denoting by a, /3, y, S the roots of the quartic X = 0, and by a, /3 , y, S f those of Y ; so that, resolved into factors, X= a(x-a)(x-/3)(x-y)(x-S), Y=A(y- a!)(y - /3 )(y - y }(y - <T) ; then A(U-aV)(U-/3 V)(U- 7 V)(U-S V) = aT\x-a)(x-fi)(x-y)(x-$} , and now a factor, such as U a V, must be composed of linear factors, such as x a, and of the squares of factors of T. In the expression y = U/V there are at most 2n + l arbitrary constants ; and in determining 7 and Vso as to satisfy relation (19) we determine 2n 2 of these arbitrary constants; thus there remain at disposal three arbitrary constants, correspond ing to the three constants involved in an arbitrary linear transformation, such as that obtained by writing ( 139) (lx+m)l(l x + m } for x, as exemplified in 153, 160, where the constants I, m, I , m are chosen so as to make X and Y quadratic functions of y? and ?/ 2 . When X and Y reduce to quadratic functions of x and y, the elliptic functions degenerate into circular and hyperbolic functions : and now there is no Theory of Transformation, except for the change from circular to hyperbolic functions, as in 16. 310 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 287. Jacobi, in his Fundamenta nova, works throughout with the differential relation for the sn function ( 35) dx connecting x = su(u, K) and y = su(u/M, X). Now, if y=U/V, then, since u = makes x = and y = 0, y and therefore U must be an odd function of x, the other, V, being an even function ; so that for an odd order of the transformation Since x = I, y = l: X = !/K, ?/ = l/X; etc., are simultaneous values of x and y, the relation connecting x and y may be written in any one of the following forms, 1+ y = (I+ x)A*/V, or V+ ff=(l+ x)A*; 1_ y = (l- x)A /z /V t V- U=(l- x)A *; (I- K x)C 2 ; ..... (22) where A and G are rational integral functions of x, of the ^(n l)th degree, which become changed into A and G when x is changed into x ; so that we may put A=P+Qx, A = P-Qx, C^P + Vx, C = P -Q x, where P, Q, P , Q are even functions of x ; and therefore l-y l^-ocP-Qx\* l-\y l^ 1 + y l+oj\P+Qaj/ l + \y l+KX\P -Q x) O^l VinO* W ~"^ $/ *"~: - a ^ a ( ^O ) When the order n of transformation is even, we put and now 7+ U = (1 + x)( 1 + Kx)B*, V+\U = D 2 , y-^=(l_a;Xl-^)5 /2 , F-X^7=D 2 ; ......... (24) where 5, D are rational integral functions of x, of the (^n l)th degree, changing into B f and D when x is changed into #; so that we may put B = R+Sx, B = R-Sx- where R, S, R t S f are even functions of x. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 31 1 288. The number of independent constants represented by the a s and 6 s in U and V can be immediately halved by noticing that a change of u into u + K i has the effect of changing x into l/ K x and y into l/\y ( 239); and therefore of interchanging U and V. An algebraical simplification is thus introduced by writing X /\/K for x and y/^/\ for y, as in 143 ; the differential rela tion now becomes of the form (Cayley, American Journal of Mathematics, vol. 9) P dx (25) and 2a = jc+l/K, 2/3 = X + l/X, .................. (26) , . sn(u, /c) sn(pu, X) . connecting x =, -, y *-~ - v K and now, if y= U/V, for an odd order n of transformation, involving only n co efficients B , B lt ... , B n _i, and therefore n l arbitrary constants in y ; also B n _i=pB . It follows then that, in the original relation y=U/V, con necting aj = sn(w, K) and y = su(u/M,\), if a 2 x 2 is a factor of 7, then 1 /c 2 a 2 x 2 must be a corresponding factor of V ; and we thus obtain the expression of y as a function of x given in equation (8), and in addition the relation \ = M*K n ILa*, (27) so that we may write y = M^xILj^_~^ (28) Professor Cayley writes equation (25) in the form (i+ where the ^ s and S s are the zonal harmonics of a and /3. 289. Writing this equation (28) in the form which is an equation of the ?ith degree in x, the roots of which are x = snu } sn(u2a>), ... , sn{u(?i 312 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. where o> = 2K/n or 2K i/n for the two real transformations, we find that the sum of the roots _ or combining the equal positive and negative values of s, \ ( u ^ \ 2 sn u en 2so> dn -^r^sm ^, X ) = sn u -h 2 = - 5 -: - 5 /elf \lf / 1 /c 2 sn 2 2so>sn 2 u the expression for ?/ when the product in equation (8) is resolved into its partial fractions ; and similar expressions hold for the en and dn functions (Jacobi, Werke, I., p. 429 ; Cayley, Elliptic Functions, p. 256). 290. We need not therefore confine ourselves, with Jacobi, to the Transformations of the sn function ; but we may some times find it preferable to seek the relations connecting x = cn(u, K) and y = cu(u/M, X), when ( 35 ; Abel, (Euvres, I., p. 363) Mdy _ dx /(! - 2/ 2 V or the relations connecting x = dn(u, K) and y = _-, 2 ~ - 2 2 ~ relations already given in (4), (5), (13), (14) of 282, 284. But Prof. Klein points out (Math. Ann., XIV., p. 116) that it is the differential form of 38 (really Rieinann s form), connecting z = sn 2 (u, /c) and t = sn 2 (u/lf, X), and leading to the relation, on writing k for /c 2 and I for X 2 , Mdt dz^ _ _ , ,oo\ ~ which is the most fundamental in the theory of the elliptic functions sn, en, and dn ; the periods now being 2/f and 2K i, instead of 4>K and ZK i, etc. ( 239) ; the quadric transforma tions (of the second order) z = x 2 , 1 x 2 , or 1 A 2 , t = y\ l-if, or 1-X 2 /, ................. (34) leading immediately to the preceding transformations of the sn, en, and dn functions. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 313 291. The Theory of Transformation may be developed en tirely from the algebraical point of view ; but Abel has shown how the form of the transformation of the nth order may be inferred from the elliptic functions of the nth parts of the periods, called by Klein, modular functions. Thus taking the first real transformation connecting in relation (33), then - -nd-*y -HA D= II(l-A;a0) 2 , ...................... (35) where a = sn*2sK/n, /3 = sv-(2s-l)K/n, arid the products extend for all integral values of s from 1 to !(-!> The form of the factors is inferred by Abel from the con sideration that (i.) where s and s are integers ; and, from equation (3), z = sn 2 2sK/n = Q, or a; (ii.) when = l, ^Y=(2s-l)A+2s A% u = (2s - l)K/n + 28 jfiT *, z = sn\2s- V)Kjn = 3 or 1; (iii.) when t = l/l t u/M=(2s-I)A + (2s -l)A i, u = (2s - 1) K/n + (2s - I)K i, z = sn*{(2s-I)K/n-K i} = l/k/3 or l/k. (iv.) when t = co , u/M= 2s A + (2s - 1) A i, z = sti 2 (2sK/n - K i) = 1/ka, or oo . Similarly the relations can be inferred connecting 3=cn 2 (tt, K ) and t = cn 2 (u/M, X), or z = du 2 (u, K) and t = cn 2 (u/M, X), not only for the first real transformation, depending on equa tion (3), but also for the second real transformation, depending on equation (11), and also for any one of the imaginary transformations of the nth order. 314 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 292. In Weierstrass s form the relation is dx connecting x = p(u; g 2 , g 3 ) and y = p(u/M; 72,73), by a relation of the form y=U/V; and this must be equivalent to relations of the form y-e a = (x-e a )A*/V, or (x-efi&IV, or (x-eJC*IV, (36) for a transformation of odd order; giving so that V must be a perfect square ; thus leading to the requisite number of equations for the determination of the arbitrary coefficients in U and V, and an equation over, which relation may be made to connect the absolute invariants J and J , and corresponds to the modular equation. For a transformation of even order, we shall have U equivalent to relations of the form 9-+ and therefore A 2 x-68 B* x-e y <7 2 si* r 5= 3* or ** ..... ( > 293. In the Weierstrassian form we determine the relation connecting x = $>(u, J) and y=p(u/M t J ). But without altering J we may write ( 196) and now, if w, M denote the real and imaginary half periods of $>(&, J) or pu, we may take w/n, u> as the periods of ip(u, J ) in the first real transformation of the nih order ; and w, af/n as the periods in the second real transformation (Felix Muller, De transformations functionum ellipticarum; Berlin, 1867). The first real transformation, of odd order n t may now be written similar to equation (30) for the sn function, and obtained in a similar manner. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 315 By integration of this equation ( 195) S ~~ n), (41) 8 = 1 where G^l* 2 p(28a>/ri) = p(2sw/n) ; ................. (42) S=l 3=1 and integrating again, log <r(u, J ) = (j^u 2 + log o-u II o-(u 2sa)/ri)(r(u + 2sco/ri), <r(u t J ) = Ce G ^o-u n <r(tt - 2sw/n)o-(u + 2so/%).j ............ (43) The constant C is determined by putting u = 0, when cru a-(u = n x (r(-2sa)/n)(r( 2sw/n) and now ,r(, JQ = e^Wn" )(7(28 ^-:^ 2 r /n+M) a=l (7-(2Sft)/7l) p2sa>/7i), ...................... (44) by formula (K) of 200. Thus, for instance, with 7i = 3, o.(u, JO = e G ^(cm)*(& U - GJ, ............... (45) where (7 1 = p|o) = ^Jw, and therefore satisfies the equation of 149 v ............... ( 46 > Denoting by 6r 2 and 6^3 the transformed values of # 2 and g B , they are found by a comparison of coefficients in the expansion of both sides of equation (44) in ascending powers of u ( 195). Thus, if J"=0, or g. 2 = Q, then 6^ = or ^ 3 ; and taking the value G l = 0, then J = 0, G 2 = Q, G B = -27g B , and <r(u; 0, -27g B ) = (<ruf&u ...................... (47) Employing the principle of Homogeneity of 196, this equation may be written (r(uiJ3) = iJ3((ru) 3 &u, ............... (48) leading by differentiation to itu> .............. ( 49 > and 3 P ("V3)= -3^+-= ~^ + ...... (50) since g 2 = 0, as in 47. 316 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. Thus, if </ 3 is positive, and <o 2 , o> 2 the real and imaginary half periods ( 62), then ft) 2 /a> 2 = i^/3 ; and if we take u = fo> 2 , then & 3 u = g 3 ( 166, 233) ; so that pw 2 / = 0. Again, putting u co 2 in equation (49) gives %V3 = 3jfe ........................... (51) Making use of the last equation of 202, we find As a numerical exercise the student may construct the following table, and also fill in the values for u = w 2 > o> 2 , Jo) 2 , -Jo> 2 , |-w 2 , |eo 2 , ... ; taking # 2 = 0, <7 3 = 1 ; these numerical results are useful in the problem of the Trajectory for the Cubic Law of Resistance, discussed in 227-234. ,+ (V2+D 3 7ry.3 IW */2 + l) 3 e T "^ 1 ^V3 -* 3 ie Linear Transformation. 294. In Chapter II. the general elliptic differential has been reduced to Legeridre s standard form and to Jacobi s, or rather Riemann s standard form (11) of 38, by various substitutions, in 39, 40, 41, 42, 43, etc., which are practical illustrations of the Linear Transformation. In 160, the six linear transformations are given which, according to Mr. R. Russell, reduce dx/^/X to the form dz/J(Atf + 6Cz*+E). In determining the linear transformations, of the form y= U/r=(aSI> + p)l(yX + 8) ................... (52) which satisfy Riemann s differential relation Mdy _ dx _-, ~ connecting x = $n\u, K) and y = su 2 (u/M,\), THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 317 we notice, by 139, that the absolute invariant J is unchanged ; so that, according to 68, there are six values of I, given by = k, 1 l-fc, 1-4; fc-1 &&gt; !-& A; and six corresponding linear transformations, in which Afi aK+bK i .(53) cK+dK i and be ad=l; (54) a, b c,d 1 1 1 1 1 1 1 1 1 1 1 1011 10 1 mod. 2. 295. But if we change to Jacobi s form by the quadric transformation, which changes x into x 2 , and y into y 2 , then Mdy _ dx _ , , " and now, forming according to 75 the invariants g. 2 , (/ 3 , A, and / of the quartic 1 x 2 . 1 /c 2 # 2 , & = 12 216 16 and .(56) lOS^l-Jfc)* Professor Klein writes >; 4 for k or /c 2 , and calls r] the Octa hedron Irrationality ; and now the absolute invariant being unaltered by a linear transformation, )41 108Z(1 -O 4 and the roots of this equation in I are found to be (57) . giving the six corresponding linear transformations of Abel ((Euvres, I, pp. 459, 568). In the reductions of Chapter II. that linear transformation has been chosen which makes k or I positive and less than unity, and also gives a real value to the multiplier M. The corresponding values of the multiplier are given by the linear transformations being, as may be verified. y=x, rfx, :r-_ + rjX* I + irj 1 + lt]X 318 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. Landeris Transformation of the Second Order. 296. The point L ( 28) in figs. 2 and 3 has been called Landens point, because of the use made of it by Landen (Phil. Trans., 1771, 1775) for his transformation, important historically as the first case investigated of the Transforma tion of Elliptic Functions, being the Quadric Transformation, or of the second degree. The ratio AD/AE being sin 2 | a or /c 2 , while EL/EA = cosa or K ; therefore, if C is the middle point of AD, LG AL-AC AE-EL-^AD GA AG _ 1 cos a \ sin 2 a _ (1 cos Ja) 2 _ 1 cos J sin 2 a The ratio LC/CA is denoted by X ; so that 21 ~ a. (l-OM */* =(l-W, and / cX / = different forms of the modular equation of the second order. Still denoting the angle ADQ in fig. 2 by 0, we denote the angle ALQ by i/r; and now ( 28) since the velocity of Q is 7i(l+/c x )^Q, perpendicular to CQ, therefore the component velocity of Q, perpendicular to LQ } LQ d^/dt = n( 1 + K )LQ cos LQG, or d\ls/dt = n(l+ K )cos LQG. sin LQG LG , , But since : = 777, = X, therefore sin I//- CQ sin LQG= X sin ^, cos ZQ<7= VC 1 - x sin V) = A(^, X) ; and d\lsldt = n(l+ K )&(\Ir, X), or ^=am{(l+ic M X} ......................... (60) Now, since the angle LQC=2(j> \fs, therefore sin(20 -<//) = A sin i/r; ................................... (61) / __ = ~ nr tan or tanx^ = \ /- f t .............................. (03) 1 K tan 2 < sin i/r = (1 H-^sin cos as in equation (92), 67. Putting nt = u, (!+K) nt = v, then sin^> = sn and we obtain the formulas (90) to (98) of 67. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 319 297. Landen starts with the relation (61) ; so that, differen tiating logarithmically, cot(20 \fs)(2d<p d\{/) = cot \lr d\[s, 2 cot(20 ^)c?0 = {cot(20 \I sin 20 d\fs sin \jr sin(20 sin 20 cosec \fs ~ cos(20 \/s) Now cos(20 - ^) = ^/(l - X 2 sin 2 x/r) = A(^, X) ; while sin 20 cot i/r cos 20 = X, cot \js = cot 20 -f X cosec 20, cosec V = 1 + (cot 20 + X cosec 20) 2 , sin 2 20 cosec 2 ^ = sin 2 20-f(cos 20 + X) 2 = 1 + 2X cos 20 + X 2 = (l+X) 2 -4Xsin 2 0, or sin 20 cosec \lr = (l+\)J(I- K 2 sin 2 0) = (1 + X)A(0, AC), where /c = 2 x /X/(l+X) ; so that, finally, c?0 _K1+W d+ _(l+QcZ0. ~ so that, if = am(n, K), then ^/r = am{(l+/c / )ni, X}, and the angle \/r may be made to represent pendulum motion on the circle CRL, on CL as diameter, LQ meeting this circle in It. The velocity of R will then be due to the level of L , a point on CE produced, such that CL = CL/X 2 ; and now we find that EL = CL -CE=EL, after reduction, so that L and L are the limiting points of the circle AQD with respect to the horizontal line through E\ but now the value of g in the motion of R on the circle CRL must, in accordance with 20, be reduced to J<?(1 O 4 - . L Q_L D_EL+ED_ K + K - 2 _I+ K LQ ~LD -EL-ED- K - K *-T^" so that ( 28) the velocity of Q is n(l+ K )LQ, or n(l- K )L Q ................. (65) The period of R in the circle CRL is half the period of Q in the circle AQD; so that, if A denotes the real quarter period of the elliptic functions of modulus X, ^ or (l+A)A = Jf. . ..(66) 320 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 298. Conversely, as in 123, we can express the elliptic functions of modulus K and argument (1 + \)v in terms of the elliptic functions of modulus X and argument v ; or starting with the motion of R, we can deduce the motion of Q. But considering the motion of Q as defining in a similar way the motion on a larger circle, to a larger modulus y, we change X into AC and K into y, where 1 y , 1 K and now, from 123, ,. . x 1 AC sn 2 (u, AC) dn(l+/c.u, y ) = - -^-M> /., v cn(l + K . w, y) = 1 + AC sn 2 (u, AC) cn(u, Ac)dn(u, AC) 1 + K sn 2 (i6, AC) called Landens Second Transformation. With x = sn(u, K ), y = sn(l+K .u,y\ v/here y then .(68) AC), and - y 2 . 1 - yV ) Or, with x = dn(u, AC), 2/ = dn(l + AC . u, AC), y = 9* = 2/c -s-F, 1-2/ =2(1 -a; 2 ) s-F, leading to the differential relation, (3) of 35, dy _ (1 + K)dx (69) (70) THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 321 299. Denoting by T the real quarter-period of the elliptic functions to modulus y, then x = 1 makes y = 1, or u = K makes (l+ K ).u = T ; so that or (66) (l + X)A = /i=4(l + y )r ......................... (71) Also, A , K , r" denoting the corresponding quarter periods to modulus X , K, y, the imaginary transformations of 238 show that, with iu = v, , N/ v cn(v, K)dn(v, K ) cn(l+,c .v, X)= -.- cn(l+/c .v, y ) = 3 /i , / >/N 1 /c sn 2 (v, ic ) dn(l + K . v, X ) = . , , z , (> l+rsn*(v, /c) 1 (1 /c)sn 2 (t , /c ) - so that A = (1+OK , r or i(l4.X)A = K / = (l + y )r ; .................. (73) and therefore 1 A! = ^ = 2^. ... (74) J A -LV 1 An inspection of Landen s formulas shows that the dn func tion has always a rational Quadric Transformation. Mr. R. Russell shows (Proc. L. M. S., XVIII.) that the general rational quadric transformations which reduce dx/JX to the form are always of the form P v P 2 , P 3 denoting the quadratic factors of G, the sextic covariant of X ( 160). Thus if X = 1 - x 2 . 1 - K*X*, the sextic covariant may be written leading to Landen s transformations, given above. G.E.F X 322 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 300. Landen s Transformation is useful, as employed by Gauss, for the numerical calculation of K ; for if we put (fig. 2) LA = a, LD = b-, and GA = a v GL = J(a* - b*) = |( - &) ; then o 1 = J(a + 6), b, = J(ab); and K = b/a, X^bja^ ...(76) Now, denoting \js by <f> v and X by K V equation (64) becomes 2^0 dfa . (?7) ^(tt 2 cos 2 + 6 2 sin 2 0) ^/(a 2 while 0! = TT, when so that i + 6 1 2 sin 2 ] _ /*" _ % y JW**fa+bfafy or = l aa fl = K 1 (l+ Kl ) ...................... (78) Continuing this process with 15 a lf and b lt so as to obtain a continuous series, given by ( 296, equation 62). = - tan <p n , , 6n+l = V( a n&n); ............. (79) then a rt and 6 W tend to equality ; so that, putting a= & =/* and =^ /!L_ J ^/(a 2 cos 2 r*r or a ._, .-., . -*) (8<>) r=l r=l Denoting the modular angle of K H by n , then Kn+l = sin (9 n +i = tan 2 J$ n ; COS 9 n+ i = SeC 2 J0 n -v/( COS ^n) and 1 +/CH+I = sec 2 J0 n = // n ^x > so that jfir= jTrsec OVC 008 cos ^ cos 2 cos 9 3 ...), (84) a formula suitable for the logarithmic calculation of K. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 323 The Transformation of the Third Order, and of higher Orders. 301. According to Jacobi s method, the transformation may be written 1 _i_<?/ i i ->-v i i ~/ v"*y connecting x = sn.(u t K) and y = sn(u:M, X) ; and then i q + 1 + aV _ x I-x*/a z l + (a 2 + 2a)B 2 ~ M l-K*aW~ so that l/Jf=2a+l, - leading to the differential relation * We shall find that, expressed in terms of a, = and ,,_(l-a)(l+a) s ,, _ ( 1 + aX_l - a) 3 "" so that leading to the Modular Equation of the Third Order. We shall also find that this transformation ma} be written 1 cn(u/M, X) _ 1 en u /a -f 1 + a en u\ 2 1 -f- cn(u/M, X) 1 + en uAa -f 1 a en uJ * l+dn(u/4fl A)~~l+dn iAa+1-dn J ^ 88) As a numerical exercise the student may work out the case of a In Legendre s notation, with ic = siu0, ?/ = sinx/r, he finds tliat these relations are equivalent to The Transformation of the Third Order was the highest to which Legendre attained, until it was pointed out by Jacobi in the Astronomische Nachrichten, No. 123, 1827, that Trans formations exist of the fourth, fifth, or any other higher order, as already explained. 324 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. tt , Thus the transformation of the fifth order may be written 1 _ x n _ and of the seventh order and so on. 302. When the transformation of the third order in 157 is employed for the reduction of the integral in equation (6), 227, then s 3 =-E" 3 /P 2 , (92) where P = p 3 - 3p 2 sin 2 a + 3p, (93) and ./T => 2 cos 2 a+psin a 1, (94) as in equation (27), 233 ; so that ^=0 and 8 = at the points of minimum velocity. Now, with this substitution of 157, s = p(yx/w 2 ; 0, A), (95) where A = 4 - 3 sin 2 a = 27# 3 , (96) ( 228) ; and denoting /ao -V<1*> then 0Q 9 = 0, p JQ 2 =-/J f A > and # 9 Q 9 = i-Tr^S ( 293). Again (157), v ( where J=p*(3 sin a - 2 sin 3 ) - 3^ 2 (2 - sin 2 ) + 3p sin a - 2, and J+P /v /A = 2{i(sina + /v /^)P- 1 } 3 /-P /v /A = 2{Ksina-v / A)p-l} 3 ............. (97) Now from 233, ^/A = cos a(tan /3 + cot {$), |(sin a 4- >v/A) = J cos a(tari a + tan /3 + cot ) 8) = cos a tan /3, J(sin a x/ A) = J cos a(tan a tan /3 cot /3) = cos a cot $ . ., sin 6 while p = -. - -- cos(a 6) Therefore (^; u> -A) -^112.2 5 0, - cos a tan ^ sin cos(a 0) _ tan(/3 0) _ tan (j> ~ cos a cot /3 sin cos(a 0) ~ tan /3 ~"tanj _V (u- 0, .g 3 )-^ ( 234) a curious result of this transformation. THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 325 Again, since $ %a). 2 = tf ^wy we may put and then, making use of relation (17) of 229, <r(ia> 2 - u)o- 2 ( ^. 2 + w)P(il2 + u) by means of (K) 200, and the relation #?f o.\ 2 = ; and this again, by equation (CC) 279 and by 293, reduces to _ fl r(}Q i - tt ; 0, -A) .. - ; 0, -A) 6 Transfoi^mation of the Theta Functions. 303. Taking the function, as defined in 263, 265 in the factorial form, 0(x,q) = <j>(q)Il(l-2q 2r - l cos2x + q* r - 2 \ ...... (100) r = l where <f>(q) is a certain function of q which 264 shows can be written <f>(q) = H(l - q- r \ .................................. (101) then changing x into nx, and q into q n , 0(nx, q n ) = (p(q n )Ii(l-2q-" -- l c II_ S f[ {l-2(/ 2r - 1 c ? = ! = o (by Cotes s Theorem of the Circle of 270) f i.( f1 n\ s = n-l = {Sl?> a!+8 /n ?) ...................... (102) Similarly, with yu = l, 2, 3, Forming the quotients, and writing x for %7rU/K, then ( 263) 1 0-.X and thence we obtain the formulas for the Transformation of the Elliptic Functions of 283. 326 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. Similar considerations will show that, when q is changed to where yu = 0, 1, 2, 3 ; this is left as an exercise (Enneper, Elliptische Functionen, 38). EXAMPLES. 1. Prove that a transformation of the fourth order is 1 1 x lKX/Ix +y and prove that the relation between X and K is then and M= 2. Prove that, by means of the substitutions cosh ^u sinh <& =-r-r , u + sinh u cosh cosh ^u sinh < or sin = . . , t rn ~i sinh JM + cosh iucos u cosh dO = sech cosh (cosh u + sinh u cosh o 1 .3. 5...2m-l 1 /* ^ (sinh ^) -3.. . 2u - 2m o 3. Prove that, with the homogeneous variables # 1} # 2 of 155, and writing X l for dX/ dx^ X 2 for dXj dx^ the general cubic transformation which reduces dx/ +JX to the form is of the form z = (lX 1 + mX 2 )/(l X l + m X 2 ) (ex. 8, p. 174). Prove also that the general quartic transformation may be written z = (lX + mH)/(l X + m H), where H denotes the Hessian of the quartic X ( 75). (R. Russell, Proc. L M. 8., vol. XVIII.) THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 327 4. Prove that (Cayley) satisfies the relation dy pdoc Modular Equations. 304. In the Transformations of the n th order, which con nect the Elliptic Functions of modulus X with those oi modulus K, and make r = q n , or q lln , or a)Pq l/n ( 285), A i K i 1 K i 2pK+K i ,, .aK+bK i nn ^ " or - or ~ or s enerally (106) where 6c ad = n, the Modular Equation, which determines X in terms oi /c, is of the (?i+l)th order, as already stated, when n is prime, and has two real and n 1 imaginary roots. We shall content ourselves with merely stating the Modular Equations of simple order, connecting K, X and AC , X , adopting the form and classification employed by Mr. R. Russell in the Proc. London Math. Society, Vol. XXT. CLASS I. 71 = 15, mod. 16 ; Q = 4 E = 4 7i. = 15, P 3 - CLASS II. w = 7, mod. 16; = 7, P = 0, or 4/(/cX)+ / 4/(/c X / )-l, (Guetzlaff). -23, P-3 = o, or 4/( / cX)+4/(/c / X / ) + (256 / cX/X / ) TV = 71, P 3 -4E4(P 2 -Q) + 2PPl-P-0. = 119, P 8 -J^(7P 5 -28P 8 Q-hl6PQ 2 )+ J R 8 (...)...=0. 328 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. CLASS III. n = 3, mod. 8 ; p= Q = J n = 3, P = 0, or J(K\) + J( K \ ) = I, (Legendre). 7i=ll, P-/# = 0, or 7i = 43, P u +...=0. 7i = .59, P 5 + ... = 0. ^ = 83, P 7 +...=0. CLASS IV. ?i = 1, mod. 4? ; w = l, P = 0. 7i = 9, P 6 - 1 4P 3 J2 + G4PQ^ - 3-R 2 = 0. 71=17, P 3 -H lop2 -64Q) + 26JAP + 12/2 = 0. 7i = 29, P^(P 2 + I7R*P- 9R%) JR^gP 2 - 64Q - 7i = 37, 71 = 53, P2{P 4 + ^(413P 3 -2 1(5 PQ)+. 305. According to Professor Klein (Proc. L. M. 8., X. ; Math. Ann., XIV.) these Modular Equations are replaced by relations between the absolute invariant J and its transformed value / , by the intermediate of quantities T and T , such that J is a certain function of T, and J the same function of T , and now, 7i = 2; /:/-!:!= (4 T -1) 3 : (T- l)(8r + l) 2 : 27r, TT =1 (60). 7^ = 3; /:,/-! :1= ( T - l)(9r-l) 3 : (27r 2 - 18 T - I) 2 : -64 T , TT=1. 71 = 4; J:,/-l:l= ( T 2 + 14r+l) 3 : ( T 3-33 T 2 -33 T H-1) 2 : 108r(l-r) 4 , THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 329 w=5; J:J-l:l= (-r- : ( T -2_ TT=125. 71 = 7; J:J-1:1 = (T 2 +13 T + 49XT 2 7()T-7) 2 : 1728 T| 72 = 13; J:J-l:l= ( : (T 2 +6T+13)(T 6 +10 T 5 +46T 4 +108T 3 +122T 2 +38T-1) 2 : 1728r. rr = 13. The Multiplication of Elliptic Functions. 306. If we perform the second real transformation upon the first real transformation, we obtain a transformation of the order n 2 , leading back again to the original modulus K ; because the first real transformation changes q into q n , and the second real transformation changes q n back again to q. We then obtain the elliptic functions of argument u/MM t =nu, since M = K/n-A, M r =A/K f in terms of the elliptic functions of argument u, by a trans formation of the order n 2 , and thus obtain the formulas for Multiplication of the argument. Thus multiplication by 2 or 3 can be obtained by two suc cessive transformations of the second or third order ; and so on. Knowing that the order of the transformation is n 2 , we infer in Abel s manner the factors of the numerator and denominator of the transformation, involving the modular functions, the elliptic functions of the 7ith part of the periods. Thus we infer, with the notation of 258, that, for an odd value of 71, snrm = UjV, ........................................ (107) where U = n sn u II IlYl - ~ v= where m, m =0, 1, 2, 3,..., $(n-l); the simultaneous zero values of m and m being excluded. as denoted by the accents, so that the number of factors is 330 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. Combining the factors by formula (7) of 137, sn(^-O/^), (108) where A is a constant factor ; and this may be written sn?w = 4nil8n(% + Q/w); ...................... (109) where m, m = 0, 1, 2, ..., +(w-l); the simultaneous zero values of m and m being now admissible. Similar considerations will show that cn?iu = nncn^-f Q/TI), ........................ (110) dnwtt=0nndn(tH-Q/tO ........................ (Ill) To determine the constant factors, change u into u + K or u+K i, when we shall find (Cayley, Elliptic Functions, 368) 4 = (-1)^-1)^2-^ B = ( K /K ^- I \ C= (l/ic )K n *- l \ By taking in 259 a rectangle OA n B n C n , in which (M 7l = ?ia > OB n = nb, and therefore containing ?i 2 elementary rectangles, we obtain a physical representation of the formulas (109), (110), (111) for Multiplication of the argument by n. Writing u/n for u, and making n indefinitely great, we deduce in a rigorous manner the doubly factorial expressions for sn u, cnu, dnu in (1), (2), (3) of 258. Again, by putting /c = or /c = l, the student may deduce as an exercise the trigonometrical formulas for the resolution of the circular and hyperbolic functions into factors. (Hobson, Trigonometry, Chap. XVII.) The Complex Multiplication of Elliptic Functions. 307. When K \K ^/D, and D is an integer, we may sup pose the multiplier n resolved, by the solution of the Pellian equation, into two complementary imaginary factors, so that and now the multiplication by n can be effected by two suc cessive multiplications by the complex multipliers a-\-ib^/D and a ib^/D, each leading to an imaginary transformation of the Tith order, not changing q or the modulus K. (Abel, (Euvres, I, p. 377 ; Jacobi, Wcrke, I., p. 489.) The first requirement then in Complex Multiplication is a knowledge of the value of K for which K jK= *JD ; and this is found by putting K = X , K = X in the corresponding Modular Equation of the order D ( 304). The equation is now, according to Abel, always solvable algebraically by radicals ; so that, returning to the question of THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 331 the pendulum in 1 5, it is possible to determine by a geometri cal construction the position of two horizontal BB , W, as in fig. 1, cutting off arcs below them, such that the period of swing from B to B is ^JD times the period from b to V. Thus the Modular Equation of the second order being written X = (1 - K )/(! +* ) we find, on putting /c = X, X 2 + 2\ = l, or \ = J 2-1, when A /A = ^/2. Putting K = \ , K=\ in the Modular Equation of the third order ( 304), 2 /v /( /c/c ) = l, or 2/c/c = i=sini7r, when K /K=J3; so that the modular angle is T W or 15. When K /K=2, /c = U/2-l) 2 (71); obtained by putting r/r = l, y = y =J^/2 in 298,299. When K /K = j5 t 2 KK = Jo-2~ 4/(2^ ) = KV 5 - 1 X or (2/c/c )~^-(2/c/c f = 1. When K IK=J*l t 2^/(^ ) = l, 2 = , 4/(2 )=i- Collections of these singular moduli required in Complex Multiplication are given by Kronecker in the Berlin Sitz.> 1857, 1862, in the Proc. L. M. S., XIX., p. 301 ; also by Kiepert in the Math. Ann., XXVI., XXXIX., and by H. Weber in his Elliptische Functioned, 1891. 308. In the expression of y = sn(a-\-ib All /D}u as a rational function of x = snu, leading to the differential relation Mdy dx , Jacobi finds (TferA;e, t. L; de multiplicatione functionum ellipticarum per quantitatem imaginariam pro certo quodam modulorum systemate) that we must restrict a to be an odd integer, and b to be an even integer ; but these restrictions disappear if we work with the en functions ; and we can even suppose that 2 and 26 are odd integers. Let us determine then the relations connecting x = cnu and ?/ = cn J( so that I/M= -J + J i leading to the differential relation dy ( where C = K/K, the cotangent of the modular angle. 332 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. If D = 4<n 1, and we denote (K+K i)/n by w , we shall then find that, when n is odd, c ic ic but, when ti is even, . -- (1 x T c c ic The arithmetical verification for the simple cases of D = 3, 7, or 15 is left as an exercise for the student (Proc. Cam. Phil. Society, Vol. V.). Formulas (112) and (113) are inferred by putting (1) y-1, when -J( - 1 + i^/D)u = 2mK + Zm K i (m + m even) ; and then u = fan K (m + m^co, x = en 2? ft). (2) ?/=-!, i -( - 1 + iJD)u = 2mK+ Zm K i (m + m odd) ; and then x = cn(2r l)w. (3) y = ic, i (w + m odd); (4) y=-ic, \(-l+iJD}n = (2m+l)K+(2m +l)K i (m+m x even): and then x= cn(2r l)co. 309. When D = 4>n + I or 1, mod. 4, the relation connecting oj = en u and y = en \ ( 1 + i^D)w cannot be rational ; but Mr. G. H. Stuart has shown (Q. J. M., Vol. XX.) that it may be written in the irrational form THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 333 where o> - (K+K i)/(2n + 1), a transformation of the order n + i ; and this is equivalent to 2 -F, F= (i_?)n(l+- -I ; ...... (114) V ic/ I cn(2?- l)o>J this is inferred in the same manner as formulas (111) and (112). For instance, with n = 0, D = 1, and K = J^/2, c = 1 ; i/ -I , -\ //-\ //^ + cn u\ en J( 1 +i)u = *J(i)J( ^ J - y \ \i en u/ equivalent to, with u = (1 -f i)v, /1 .. .1 icn z v cn(l \)v = ^^-:. ,- 1 + i cn 2 t With w = l, D = 5, 2/c/c = x / 5 - 2 ^ = /c _v/5 + l, U5 + 1. V c 2 V 2 and en i - 1 + iou where a = en i(lf+ K i). 310. Generally in the expression of y = $ulM as a function of # = #?&, where w /w or K i/K=J(-D), and the multiplier 1/Jlf is complex, of the form it is convenient to consider four classes of 1). Class A, D = 3, mod. 8 ; Class B, D = 7, mod. 8; Class C, D = l,mod. 4; Class D, D = 2, mod. 4; the class for D = 0, mod. 4, not requiring separate consideration. It is convenient also to consider the discriminant D ( 53) as negative ; a change to a positive discriminant being effected by the method of 59 ; now w Jw. 2 = i^/D. We can also normalize the integrals ( 196, 252) by taking g* - 27# 3 2 = - 1, so that g, = J/( - J). 334 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. CLASS A. D = 3, mod. 8 = 8p + 3 or 4>n-l, if n = 2 The relation connecting x and y can be written in one of the three equivalent forms -eJ IL {x-^ 1 + 2ra) 3 M)} 2 - F, V= H{x-p(2r leading to the differential relation Mdy _ dx This verifies in the particular case of p = 0, when and then This is the simplest case of Complex Multiplication, meationed in 196, and employed in 227 in the determina tion of the Trajectory for the cubic law of resistance. The form of the general transformation is inferred from the consideration of the series of values of u which make y or <p(ujM) e v e z , e%, and GO . (i.) When y = e lt u/M= 2n so that ic or $>u = e x or (ii.) When y = e 2 , = e, or THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 335 (iii.) When y = e s , u= g <pu = e v or ^w 2 4-r ft) 3 ?i or (iv.) When y=ac , it = 2gw 2 + 2 / and $m = j0(2ro> 3 /?i). Hence the form of the Transformation is inferred. By addition, we find n>n A , 7> n - 1 i A fii - 2 j re) ^O ~~ X1 1 iXy ~f XI ,-)>Aj . . ^ i ji I ^ _ _ ^ __ (o;P-(7 1 ^- 1 + (?^- 2 ...) 2 where ?i = 2p + l; and we shall find that xl 1 = 2(r 1 ; and the xl s and (? s are symmetrical functions of e v e. 2> e%, and there fore functions of g< # 3 or J\ while ^ has the same significa tion as in 293. By employing the Modular Equations given above, or employing Hermite s results (Theorie des equations modu- laires), we find =3, J=0, 9t = o, n T -* 8 =11, J=-, g,= g= ) A _^ , . /i -"-2 -jo .3~ = 3, 9s these values of J. , A 3 , A, A 5 were calculated by Rev. J. Chevallier, Fellow of New College, Oxford, who has also verified the case of D = 1 1 . D =27, J=-2 9 x5 3 --3 2 , etc. D =35, & = f /v /5{iU/ D =43, /=-2x5, ^ = 3x7x^43 (Hermite). 73^43, etc. =51, /=-64.(5 + Vl7) 3 U/ l7 + 4 ) 2 (Kiepert). 336 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. D =67, ,/=-2 9 x5 3 xll 3 , # 2 = #, = 7x31x^/67 (Hermite). D =163, /=-2 12 x5 3 x23 3 x29 3 , J(g 2 + l) = % x 7 x 11, (/ 3 = 7xllxl9xl27x /163 (Hermite). CLASS B. D = 7 t mod. 8 = 8^ + 7 = 4^-1, if n = 2p + 2. The relations connecting y==p(u/M) and o; = pu, where are found, in a manner similar to that employed in Class A ; - ej(x - e 9 ) Yl \x - p( 2 + 2n*> : } 2 -f- F, 2 4- 7, r-O Jf 2 nf {^ - pfo + 2ro,} 2 -j- F, r As simple numerical applications, /> = 1 5, ^//c/c = sin 18 ( Joubert). In these cases the Jacobian notation is almost more simple, as given in 308. CLASS C. D = 1, mod. 4 = 4n + 1. The relations connecting x = <pu and y = tfu/M), where cannot now be rational ; but, according to Mr. G. H. Stuart, we can express the relations in the irrational form a relation which may be said to be of the order n + \ and this is equivalent to THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 337 CLASS D. D an even number. In this class the simplest function to employ is the sn func tion ; for instance, with K \K= ^/2, then K = v /2 - 1 ; sn-u ~~" 2 i) " and where <o = J( K iK ) ; leading to the equations 1 K.y _ 1 #/! + K x sn wV 2 1 -f Ky 1 + 5\1 /ca? sn o>/ connecting ic = snu and y = su(l-t- / 2 i Also = 2 = ^ These transformations show that it is not possible to express cn(l-M x /2)i6 in terms of cn-u-, or dn(l + ? v /2)u in terms of u, by a rational transformation. With K /K=2, then /c = ( v /2-l) 2 (71), and the relation connecting x = snu and y = sn(l + 2i)u may be written / /y.2 \ / ^2 \ sn*4w/ (l-A 2 3n 2 2a))(l-/c- --> w here o> = i( A" iK ) ; equivalent to the relations / T \ 7 / V n - ^_\ /ij a l-y 1-lrtBJ 1- sn z so that cn(l + 2i)i6 has a factor dnw, and dn(l + 2/)w, has a factor en u. G.E.F. Y 338 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. When K /K=J6, then K = (*J3- J 2)(2- and the corresponding relation between snu and to be written down is left as an exercise. (Proc. Gam. Phil. Soc., Vols. IV., V.) It can also be shown, in the preceding manner, that the relation connecting x = <pu and y = $(ujM) where i\}\d D is an even number 2m, can be expressed by the relations -c, A = M \x-,g H -pi--<>2 / + v > As numerical exercises, we may take (i.) D = 2, when ^ = 30, $r 8 = 28, ^=- (ii.) D = 4, when </ 2 = ll, $r 8 = 7, G^- oil. In conclusion we may quote from Schwarz some general remarks on doubly periodic functions. Every analytic function </>u of a single variable u for which an algebraical relation connects <f)(u + v) with (fru and </>?; is said to have an Algebraical Addition Theorem ; and then <f>u must be an algebraical function of 0u (Chap. V.). Every such function is then an algebraical function, or an exponential function (circular or hyperbolic function), or an elliptic function, which can be expressed rationally by pu and V u (Chap. VII.). Elliptic functions are doubly periodic. A function of a single variable cannot have more than two distinct periods, one real and one imaginary, or both complex. For if a third period was possible, the three sets of period parallelograms obtained by taking the periods in pairs would reach every point of the plane, so that the function would have the same value at all points of the plane, and would therefore reduce to a constant (Bertrand, Calcid int&jral, p. 602). THK TRANSFORMATION OK KLLIPTIC FUNCTIONS. Abel, in generalising these theorems, was led to the discovery of the hyperelliptio and Abelian functions. Thus if X in 169 is of the fifth or sixth degree, we obtain functions of 2 variables and 4- periods ; if of the 7th or 8th degree, of 3 variables and 6 periods; and generally, if X is of the degree 2p + l or 2p + 2, there are p variables and 2p periods ; but this would lead us beyond the scope of the present treatise, and the reader who wishes to follow up this development is recommended to study Professor Klein s articles u Hyperdllptische Sigmafunctionen" Math. Ann., XXVII., XXXIH., etc. APPENDIX. I. The Apsidal Angle in the small oscillations of a Top. The expression given by Bravais in Note VII. of Lagrange s Mdcanique analytique, t, II, p. 352, for the apsidal angle in the small oscillations of a Spherical Pendulum about its lowest position is readily extended to the more general case of the Top or Gyrostat, if we employ the expression on p. 201, 242, as the basis of our approximation. We divide the apsidal angle ^ into two parts, ^ and "^ such that i ^ l = ar] l a) l ^(ij ^ f2 =bt h -w^b- and now put a = o) 3 sa> 3 , b = w l + qcD s , where q and s are small numbers; so that, expanding by Taylor s Theorem as far as the first powers of q and 8, we may put ^a and now, by means of Legendre s relation of p. 209, But, from equation (B), 51, vu e;, - ^ = 2 = 1 so that, integrating between the limits and oo v o or ^ + e l ta l = V( e i ~ s)^ (Sch warz, 29). Also (51) (^-^3)0)!- ^/(^-e.^K- so that i^! + <? 3 wi = V(^i "" e z>(K ~~ ty and therefore i^ = \i-jr + ^ 3 ^/( fl i 340 APPENDIX. 341 But, from 210, when a and ft are very nearly TT, their approximate values are given by since f // a = 2(e? 1 e 3 )(e 2 e s)> and K 2 = ^A ^ = Vzi2( 52 ); and therefore Also ( 210) G-Cr ; /C 2 so that (^ - e 3 )q V - - T -2 cot2 i cot 2 2 _ Therefore ^ ^ JTT + 3 jc cot Ja cot E But, ultimately, when /c = and /c r =l, then ^=i7r, and lt(/i _^)/ AC 2 = i 7r (11,170); so that Mfj ^ JTT + ITT cot Ja cot i/3, 7r cot - a cot 3. This reduces for the Spherical Pendulum, in which (7? = 0, to ^ S JTT( 1 + f cot Ja cot J/3) S $ir(l + 1 sin a sin /3), when a and ft are nearly TT, thus agreeing with Bravais s result. When a = 7r and 6r-f-(7r = 0, this approximation fails; but the student may now prove that the apsidal angle is This will be the apsidal angle when the Top is spinning in the vertical position with small angular velocity / , and is then struck with a slight horizontal blow. 342 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. II. The Motion of a Solid of Revolution in infinite friction- less liquid. The reductions of the Elliptic Integral of the Third Kind in 282 in consequence of the relation a + b = u> a , in connexion with the Top and Spherical Pendulum, are useful also in constructing degenerate cases of the motion of a Solid of Revolution in infinite liquid, as mentioned in 211. We refer to Basset s Hydrodynamics, Vol. I., Chapters VIII., IX., and Appendix III., also to Halphen s Fonctions elliptiques, II., Chap. IV., for an explanation of the notation ; and now T the kinetic energy of the system due to the component velocities u, v, w of the centre of the body along rectangular axes OA, OB, 00, fixed in the body, 00 being the axis of figure, and to component angular velocities p, q,r about OA, OB, 00 is given by T=$P(u* + v*)+$Rw z + } 2 A(p* + q*) + $Cr* (A) (to which the terms P (up + vq)+P wr may be added in the case of a body like a four-bladed screw propeller, or like a rifled projectile provided with studs or spiral convolutions on the exterior). Then the Hamiltonian equations of motion are d VT - T VT VT v m dt d Vu VT VT VT _i_ Y __ -y, - ...(2) dt d Vv ^ Q VT Vu VT -z, .. ...(3) dt d Viv " VT Vu VT VT v L dt d Vp VT VT VT VT W -M dt d Vq P VT a VT T Vp VT 4-99 VT Vu +**=*. . (6) dt Vr q Vp +r> -d<j Vu Vv When no forces act, so that X, Y, Z, L, M, N vanish, then equation (6) shows that Or or r is constant. Multiplying equations (1) to (6) by u, v, w, p, q, r in order, adding and integrating, shows that T in (A) is constant. APPENDIX. :U:{ Multiplying (1), (2), (3) by ||, g, g adding and in- tegrating, proves that /32V , /37V /32V . I - - ) + ( ) + ( - ) is constant ; or \du/ \dv J \dwJ pi(u* + v~) + RW = F 2 , ..................... (B) F being a constant, representing the resultant linear momentum of the system. Similarly, it is shown that -dT^T -dT^dT 3T3T. - is constant : or r=G, ................... (C) where G is a constant, representing the resultant angular momentum of the system. From equations (A) and (B), A( p* + q-) = 2T- 6V 2 - Ru A - P(u 2 + v 2 ) and, from equation (3), so that ^y or ^^(; is an elliptic function of t. Taking the axis Oz in the direction of the resultant impulse F, and denoting by y p y 2 , y 3 the cosines of the angles between Oz and OA t OB, 00, so that then, with Euler s coordinate angles 6, </>, ^, = sin $cos = sin^sin -_p cos + sn ~ so that 344 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. G + CFr J__ G-CFr 1 4 " __ 2AF 1+C080 2AF l-cos0~ dt suppose ; and then , CFr-GcosO The equations given by Kirchhoff ( Vorlesungen iiber mathe- matische PhysiJc, p. 240) for a, fa y, the coordinates of with respect to fixed axes O a, /3, O y (O y parallel to Oz) are ^- - ^-; .................. (9) at Vu &v 3w where a v 2 > a s denote the cosines of the angles between O a and OA t OB, 00; and fa, fa, fa, the cosines of the angles between O p and OA, OB, 00. Expressed b}^ Euler s coordinate angles, a^ cos 6 cos <p cos \^r sin <f> sin ^, 2 = cos sin cos \/r cos (/> sin i/r, a 8 = sin cos ^r ; fa = cos cos sin i/r + sin cos \/r, fa = cos sin sin i/r + cos cos i/r, j# 3 = sin ^ sin i/r ; while p = s ^ n <f>0 sin cos \^, g = cos 00 + sin sin i/r, r = 0-}- cos \js ; so that, after reduction, Fa = A cos i/r ft + (Or J. cos ^)sin 6 sin i/r, ^S = A sin \/r - (Or - A cos ^)sin 9 cos ^, Writing Fx for ^cos or Rw, equation (D) becomes n *i(x> if (IT rv ^V 2 - 1 <& - l (a " Tj/ 1 4 2 suppose, where n 2 T\D~pJ- APPENDIX. 345 Denoting the roots of the quartic X = by x ot x v x 2 , x^ we may put, according to 151, 152, x-x = j. _ / - . ._ _ 3 pu pc pc e B and now, when x oscillates between x. 2 and x%, u = nt + o) 3 . The letter u has been used here in two senses, to agree with the ordinary notation ; this need not however lead to confusion. Differentiating, (u f> 2 2 <p ( u c) ^ 2c - c) + p(-u -f c) ; so that we must write v for 2c and u for u c, to agree with Halphen s notation. Now, to determine y, F = (F- ^ nP so that, in a complete period 2^ of the motion, the point will have advanced parallel to O y a distance F 2 also ( 152) 6^2c = coefficient of x 1 in X. 346 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. We now suppose that u = a makes #=1, and u = b makes x 1; then (pa - <pc)(<pu - pc) (p& - J?^Lf?l c _ _ .G CFr p 6 p c _ .6 + CFr (pa - pc) 2 ~ 4Jfy, " Then di\ls l _ \ p a(pit- </% (pa ^ ! pa pc pM,-a = - !(( ~ c ) - KO + c) + fa and similarly ; ^i=- and therefore ty = - .|Pu + i log where Also = -a; = $> 2 c($>u - pa)(p?> - pu) (pa - pc)(p6 - pc)(pw - p o-(a - c)a-(a + c )o-(6 - c)o-(6 + c)a- 2 (u - so that giving the projection on a plane perpendicular to 00 of the motion of a point on the axis OC t relatively to 0; also We find also, as in 224, that if the values a x and b l of u correspond to then APPENDIX. 347 But now introduce the condition a + b = w (t , when, according to 282, \[s becomes pseudo-elliptic. Putting =tan- and, employing 6 instead of a, this may also be written so that 1+X Q . l + X a l-X Q . l- and therefore each is equal to 1, and snce x + and, changing to the complementary angle. XR.X X-y X X Q .XXa -sin-i /^-^-*7_ M -i Ix-Vo-Xc^x V 2^2^"" V ~2-7 2 with o. a > a;^ > a; > ic 7 > # . Differentiatin, so that -I ~~ iC" provided that n(x +x a ) = GjAF, n(l + x x a ) = Cr/A. The quartic ^ must therefore break up into the two- , ,. fe (7?- Gx Cr , quadratics ^-- + -1 and ^+-^--,--1$ Mid = (T&_W_( GX - CrF ^ so that the requisite relation when a + 6 = o>, is 348 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. Now so that sin 2 # sin 2= ^A r , sin 2 cos 2= and g=mt-\ls, where m = J (a; + a; a ) - J G/A F. Also, from (7) and (8), F(a cos r + 3 sin \/=A6 = A n^/JT/sin 6 = An sin sin 2f ; JP( sin \/r /3 cos i/r) = (Cr xl cos 0\^)sin ^ CrF-GcozO -f^TO -An sm0 cos 2 Therefore ^ a = A n sin 0(sin 2 cos \[s cos 2 sin \/r) = J.?i sin sin(2 \js) = Ans\n sin2m#- ^/3 = ^1% sin Q c = ^4 sin 6 c Now in the motion of a point on 00, relative to 0, sin 0e^ = sin cos(mt g) + i sin 0sin(m _^r/ jX-X Q .Xa-X . lxp-X.X W" ~"2~ *V~ ^~T~ where a; = cos a When b a = wa t and V^i"^ or ^ s pseudo-elliptic, we shall find that 6r and Or are interchanged, and JCtJ then 2T-Cr z - ^ = ; ........................... (F) ^L so that P*(V + v 2 ) = As a numerical exercise, we may take, in addition to (F), G = 4>AFn, Or =2^/7 An; then X = x 4 -Wx 2 +IG / J7x-l5 = (a; 2 - 2J7x + S)(x* + 2 ^ - 5) ; = 60, (/ 8 = 88, ^ = 1+2^/3, 2 =-i>, 3 = 1-2^/3; APPENDIX. 34,9 pa = - 8, pb = 1 ; a = a> 8 , 6 = Wl - Ja> 3 ( 225) ; pc = 2 v /7 + 3, p c = - 8^7 - 20, 9 2c = 5, p 2c = Now we shall find that =etc. sin 3 cos 3(nt - \fs} = ( - f + *J7 cos - cos 2 0)*, sin*08in3(nt ^) -2 cos + 7 cos/- cos 0- MISCELLANEOUS EXA^IPLES. 1. Construct a Table exhibiting the connexion between th^ twelve elliptic functions sn u, ns u, dc u, cd it ; en u, ds u, nc u, sd u ; dn it, cs u, sc u, nd u. 2. Construct a Table of the values of the sn, en, dn of u-\-mK-\-nK i in terms of sn u, en it, dn u ; also of the elliptic functions of ^(yiiiK-\-nK i), for m, tl=0, 1, 2, 3. Prove that, accents denoting differentiation, (i.) sn it dn"it sn"u dn u = sn u dn u, etc. (suit) 2 , snt&sn tt, (sn u) 2 (ii.) |(cnu) 2 , en it cn%, (cn tt) 2 = /c /2 sn u cuudn u. (dnu) 2 , dnudn fi, (dn^t (G. B. Mathews.) 4. Denoting by (m, n) the function sn (u m u n )cr\(u m + u n ) en u m prove that (4, 1)(4, 2)(4, 3X2, 3)(3, 1)(1, 2) + (4, 1)(2, 3) + (4, 2)(3, Denoting by A, B, G the functions z] sn( 7/)sn(s a;) z)sn(x y}" prove that ABC+A + B+C=0. 350 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 5. Prove that /~2u (i-) IK sn vdv = 2 tanh - l ( K sn 2 u). o (ii.) A sn(2 it -f a)du = tanh ~ x { K sn u sn(i& -f a) } . o fK (ill) / log us mfot = \TrK -\K log I/*. o 6. Determine the orbit in which P = h?(u 3 +a 2 u 5 ), the apsidal distance being a. 7. Rectify r*a*co8|0, 8. Prove that the perimeter of the Cassinian Oval of 161 . 4/3 2 # IB e^her - = and draw the corresponding curves. 9. Prove that the length of the curve of intersection of two circular cylinders, of radius a and &, whose axes intersect at right angles, is 8a %l **^ K 2 = 2 /6 2 ; and verify the result when a = b. 10. Prove that K and K satisfy the differential equation d ^n fr^l 1 7^ o d/c\ L(l ~ Lj dkr 4 Deduce the relation die and thence deduce Legendre s relation ( 171). 11. Prove that CTJ and C7 2 of 252 satisfy the differential 2 - equation 12. Deduce the Fourier series for snu, en u, dnit of 266, 267 from the series for Zu of 268, making use of Landen s Transformations and of equations (28), (29), (30) of 264. MISCELLANEOUS EXAMPLES. 351 13. Prove that n i s _ ; .... , 14. Prove that, if a variable straight line meets the curve Atf+Byt+Cx+D^Q , y 2 X^s y a ) then ( 166 ) .Vl 2/2 ?/3 15. Denoting the integral /* xydx , ~ tf^x by fo) o where y is given as a function of # by the equation prove that, for three collinear points, 16. Prove or verify that, with </ 2 = 0, the solution of Lame"s differential equation s = 1 s = (Halphen, Memoire sur la reduction des equations difftren- tidles, 1884.) 17. Determine, by means of elliptic functions, the motion of liquid filling a rectangular box, due to component angular velocities about axes through the centre parallel to the edges. (Q. J. M., XV., p. 144 ; W. M. Hicks, FeZocify and Potentials betiveen parallel planes, p. 274.) 352 THK APPLICATIONS OF ELLIPTIC FUNCTIONS. 18. Prove that, with x = %7ru/w and A = ^/w ( 278), and thence convert the formulas (M) to (T) of 249 into- Jacobi s notation. 19. Prove that ( 264, 20*) l/Q = T fl(l _ fr) = m r=l m 20. Prove that (in.) ,_= II- 21. Prove that, in Appendix II., p. 34G, O 2 - b} - Work out the case of An* INDEX. PAGE Abel, - 4, 18, 145, 223, 277 Abel s theorem, - 166, 249 pseudo- elliptic integrals, - 2*28 Abelian functions, - - - 175 Addition theorem, of circular and hyperbolic functions,- - - - 112 for elliptic functions, - 113, 117 of second kind, 178, 180, 193, 226 of third kind, - 193, 224 algebraical form of, - - 142 Theta function, - - - 192 Allegret, - -, - 147 Amplitude, - 278, 289 hyperbolic, - - " - - 15 of elliptic integral, - 4 Anharmonic ratio, of four points, 53, 57 Anomaly, mean, of a planet, - 14 Apsidal angle, - - 260, 340 Argand, - - - - - 46 Argument, - 191 Ballistic pendulum, Xavez, - 3, 12 Bartholomew Price, 3 Basset, . . 288 Basset s Hydrodynamics, 219, 342 Bertrand, - - - - 9 Biermann, - - 77, 151 Binet, - - - - 213 Bjerknes, - . - 22 Boys, ... . . 97 Bravais, - - - - 340 Brioschi, - - - - - 275 G.E.F. i Burnside, W., - and Panton, PAGE - 38, 107, 172, 209 148, 150 Capacity, coefficients of electric, 287 Capillary attraction, - - 89 Cartesian ovals, - - 257, 262 confocal orthogonal, - - 255 Cassinian oval, area of, . 189 rectification of, - - 164 Catenaries, - - - - 76, 92 of uniform strength, - - 92 Catenoid, - - - - 95, 98 Cauchy s residue, - - - 206 Cayley, 56, 62, 139, 142, 160, 280, 311, 327 Central orbits, ... 76 Chain, revolving, - - 67, 210 Chasles, - - - 178 Chevallier, - - - - 335 Chrystal, .... 66, 277 Circular functions, trigono metrical, - ... 6 Clifford, - - - 17, 30, 284, 295 Complementary modulus, - 9 Conductivity, thermometric, - 285 Confocal, ellipses and hyper bolas, - - - 184, 255 quadric surfaces, - - 271 paraboloids, - - - 273 Cotes s spirals, 75, 190 theorems, - - 289, 325 Cubic substitution, 41 353 354 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. Cubicovariant, Cubin variant, - Cycloidal oscillation, Cycloids, - PAGE 158 62 7 190 Darboux, - - 221, 236, 255 Delambre, - 137 De Moivre s theorem, - - 288 De Sparre, - 233 Despeyrous, - 236 Descriminant, - - 44, 48, 158 Discriminating cubic, - - 154 Doubly periodic function, 208, 254, 299 Duplication formulas, - - 120 Durege, - 117,227 Dynamical problem, - - 74 Elastica, - 87, 190 tortuous, - - - 213 Electrification of two insulated spheres, - - 287 Electrode, - - - - 278 Ellipse, rolling, - - - 71 first negative pedal of, 73 Elliptic functions, addition theorem, - - 142 complex multiplication of, 12, 203, 330 double periodicity of, - - 254 geometrical applications to spherical trigonometry, - 131 multiplication of, - - 329 reciprocal modulus, - - 24 resolution of, into factors and series, - - 277 Elliptic integral, of first kind, - - 4, 22, 30 of second kind, - 64, 175, 209 of third kind, 108, 175, 191, 206, 302 complete, - - 8 definition of, 5 degenerate, - - - 41, 57 factor of third kind, - - 226 general, - 200 graphs of, - - - - 66 half period of, - ... 13 Elliptic integral coiitinued, inversion of, 30 modulus complementary, - 9 normalised, - - - 203 quarter period of, - 8, 321 quarter period, complemen tary, 9 Tables of, - - 10, 11, 16, 177 Weierstrass s defined, - - 42 Enneper, - - - 61, 326 Epitrochoid, - 190 Eta function, - - 194,282 Euler, - - - 142, 251 Euler s addition equation, 144, 166 constant, - - 281 equations of motion, - 18, 101 pendulum, - > 198 Fagnano s theorems, - 182 Forsyth, - - - 298 Fourier, ... 66 series, 285, 287 Fricke, - 155 Fundamenta Nova, - - - 310 Fuss, 121 Gauss, - - , - , 137, 322 Gebbia, - - - - - 220 Genocchi s theorem, - 264 Geodesies, .... 95 Glaisher, - 17, 33, 62, 116, 133, 194 Governor, Watt s, - - 78 Graphs of elliptic integrals, - 66 Graves, ... - 178 Gudermann, 5, 32, 90 Gudermannian, - . 14 Half period, imaginary, - - 44, 50 - real, 43, 50 Halphen, 128, 130, 206, 217, 276, 342 Hammond, - - 256, 294 Harmonic motion, 13 Heat, conduction of, 284 Helicoid, - - . - - 95 Helix, ..... 20 Hermite, 150, 158, 208, 215, 276, 335 Herpolhode, - 101, 107, 207, 231 algebraical, . . . 228 INDEX. 355 62, Herpolhode, points of inflexion, Hess, Hessian, - Hicks, Hill, ... Hobson, - - 277, Holzmiiller, - Homogeneity, - - 203, Homogeneous variables, - Hooke s law, - Hoyer, - - - - Huygens, Hyperelliptic function, - integral, Hyperbolic amplitude, - functions, - Hypotrochoid, - 233 - 233 149, 156 288, 351 291 280, 330 - 257 247, 270 - 155 94 237 6 - 175 168, 309 15 15 - 190 Icosahedron form, - - 156 Imaginary period, - - - 254 Induction, electric coefficient, 287 Inflexion, points of, on herpol- hodes, - - - - 233 Integrals, circular and hyperbolic, - 30 hyperelliptic, - - 160 poles of, - 45, 53 Invariants, - - - 43, 62, 143 absolute, - Jacobi, Jacobi s notation, Jenkins, - Jochmann, - 45, 49, 143 - 5, 139, 160, 284 - 18, 50 84, 131 - 278 Kaleidoscope, - 293 Kepler s problem, 14 Kiepert, - 331 Kirchoff, - - - 87, 344 Kirchoff s kinetic analogue, - 214 Kleiber, - 190 Klein, - 35, 151, 271 Kronecker, - ... 146 Kummell, - 136 Lagrange, - 131, 340 Lambert s series, - - - 287 Lame s differential equations 210, parameters, Landen s point, - - transformation, - 55, 60, second transformation, Lecornu, Legendre, - 4, 18, 64, Legendre s relation, Lemniscate, ... rectification of, - - 15, Linear substitution, transformations, - Lintearia, Lodge, Love, - - - 216, 275 272, 274 23, 117 186, 322 120, 320 - 212 131, 323 164, 178 . 199 - 33 - 190 - 143 163, 316 87 278, 293 - 293 MacCullogh, - MacMahon, Mannheim, Maxwell, Mean anomaly, Mercator s chart, Sumner lines, Meridional part, Michell, - Minding, - Modular angle, equations, - equation of third order, Modulus of elliptic integral, - change f rom , and its reciprocal, complementary, - singular, 179, 220 - 147, 295 - 221 79, 89, 272, 287 14 17 89 17 - 292 . 121 4 - 323, 327 323 4 24 9 331 Morgan Jenkins, - 84, 131 Motion, of a body in infinite liquid under no forces, - 219, 342 of a projectile, resisting medium, 65 of electricity or fluid, - - 266 mean, of a planet, - 14 Poinsot s geometrical repre sentation, .-- - 101 solutions of Euler s equa tions of, - - 28, 101 Miiller, 314 356 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. PAGE Napier, - 137 Nodoid, 95, 98 Norm, - - - ....- - 278 Octahedron form, - 157 irrationality, - 317 Orbits, central, . - . . 76 Oscillations, cycloidal, 7 quadrantal, - - 103 rectilinear, - - - 25 of pendulums, bell, etc., - 3 vertical, of a carriage or ship, 82 Parameter, 191, 207 Pendulum, 1 Euler s, - - 198 - Navez, ballistic, - - 3, 12 performing complete revolu tions, - 18 period of, 8 reaction of axis of suspension, 82 simple equivalent, - - 3 speed of, - 3 spherical, 214 Period, parallelogram, 46 rectangle, - 270 Poinsot, - - 233 Poinsot s geometrical repre sentation of motion, - - 101 Poles of integral, - - 255 Polhode, - - 101 separating, - 230 Poristic polygons, Poncelet s, - 121 heptagons, - 130 pentagons, - - 128 quadrilaterals, 126 triangles, - 124 Poundal, - 1 Price, - 3, 79 Pringsheim, 160 Projectile, trajectory of, for cubic law of resistance, - 244 Pseudo-elliptic, 242, 347 integrals, Abel s, .- 228, 300 Quadrantal oscillations, - - 103 Quadri-quadric function, - - 148 PAGE Quadric surfaces, confocal, - 271 transformations, - - 35, 321 Quadrinvariant, 62 Radian, - 1 Real half period, - 43 Reciprocant, - ... 294 Reduction, formula of, - - 63 Reversion of series, - - - 202 Revolving chain, - - 67, 210 Richelot, - 121, 164 Riemann, 312 differential relation, - - 316 Roberts, - 162, 165, 199 Robertson Smith, - 291 Rocking stone, - .- - 198 Rolling and sliding cone, 108 Routh, - 3, 28, 101, 217, 238 Russel, - 62, 149, 151, 158, 327 Salmon, - 149, 157, 162, 178, 222 Schwarz, - - 26, 46, 157, 298 Sextic covariant, 150, 157, 163, 321 Siacci, - - 220 Sigma function, - . - 201 Simple harmonic motion, - 13 Simpson, - - - - . - 9 Slade, - - .. .. 233 Smith, - 27, 222 Spherical pendulum, - - 214 Spherical trigonometry, - - 169 geometrical application of elliptic functions to, - 131 Spinning top, - - - . - 214 Spiral, Cotes s, 75 Steiner, - 121 Substitution, linear, 143 Sumner lines on Mercators chart, 89 Surface, special minimum, - 26 Swinging body, internal stresses of, - 84 Sylvester, 221, 294 Syzygy, - 150, 156 Tables of elliptic functions, 10, 11, 16, 177 Tait, - - - .- .. .;-. 87 INDEX. 357 Talbot s curve, Temperature, stationary, Theta function, addition theorem for trans formation of, Thomson, Sir W. , - Thomson, J. 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